This document discusses a report on applying probability to control traffic flow through a highway system. It first defines key probability terms like events, sample space, mutually exclusive events, and independent events. It then provides an example calculation of probability by finding the probability of an even sum when rolling two dice. Finally, it describes a proposed probabilistic model for modeling lane changing behavior on highways based on drivers' desired speeds and safety constraints like the two-second rule for following distances. The aim is to provide a simple yet accurate representation of lane changing behavior to help control overall traffic flow.
This document discusses conditional probability and independence. It defines conditional probability as P(B|A), the probability of event B given that event A has occurred. To calculate this, one considers only the outcomes where A occurred and calculates the fraction where B also occurred. Two events A and B are independent if P(B|A) = P(B), meaning the probability of B is unaffected by the occurrence of A. The general multiplication rule for any events A and B is P(A and B) = P(A) × P(B|A) or P(A) × P(B|A). Disjoint events cannot be independent as the occurrence of one rules out the other. Conditional probabilities are best understood
Conditional probability is the probability of an event occurring given that another event has occurred. It is calculated as the probability of both events occurring divided by the probability of the first event. An example is given of calculating the probability of drawing two white balls in succession from an urn without replacement. The formula for conditional probability is derived as the probability of events A and B occurring divided by the probability of A. This is demonstrated using an example of finding the percentage of friends who like chocolate that also like strawberry.
These slides represent a brief idea about conditional probability along with illustrative examples and discussions. It also consists the use of sets to develop a better understanding for the students having the following theorem in their course.
The document discusses smoking-related deaths in the United States each year, including 123,800 from lung cancer out of a total of 438,000 smoking-related deaths. It then defines conditional probability as the probability of one event occurring given that another event has already occurred. Using this definition, it calculates the conditional probability that a smoking-related death will be caused by lung cancer as 28%, based on the number of lung cancer deaths divided by the total number of smoking-related deaths.
This document discusses key concepts in probability. It defines basic terms like experiment, sample space, event, and probability. It provides examples of calculating probability for coin tosses and dice rolls using the classical method of dividing the number of ways an event can occur by the total number of possible outcomes. The document also discusses limitations of the classical method and introduces the empirical and subjective methods of determining probability based on observed frequencies and personal judgment respectively.
Discrete and continuous random variables can be used in various engineering applications. Discrete random variables take on countable values and are used when things are counted, like the number of defective items in a batch. Continuous random variables can take any real number value and are used when measurements are made, like the time for a chemical reaction. Some examples given include using discrete variables to find beam loading at points or quality control sampling, and continuous variables to estimate construction time, structural load magnitude, electrical current amounts, and component failure times.
1. Mathematical expectation, or expected value, of a random variable is calculated by taking the sum of each possible outcome value multiplied by its probability of occurring.
2. For a game with multiple outcomes and their associated probabilities and payoffs, the expected value is the average amount a player can expect to win or lose over many plays. A positive expected value means the game favors the player, while a negative value means it is biased against the player.
3. Variance measures the average amount that values of a random variable deviate from the mean or expected value. Unfortunately, the document does not provide the formulas for variance or standard deviation of a random variable.
This document provides an overview of probability concepts including:
- The three axioms of probability: probabilities are between 0 and 1, the probability of the sample space is 1, and the probability of the union of disjoint events equals the sum of the individual probabilities.
- Formulas for probability, conditional probability, independence, and complements.
- Discrete and continuous random variables and their properties including expected value and variance.
- Examples of probability mass functions for binomial and Poisson distributions.
This document discusses conditional probability and independence. It defines conditional probability as P(B|A), the probability of event B given that event A has occurred. To calculate this, one considers only the outcomes where A occurred and calculates the fraction where B also occurred. Two events A and B are independent if P(B|A) = P(B), meaning the probability of B is unaffected by the occurrence of A. The general multiplication rule for any events A and B is P(A and B) = P(A) × P(B|A) or P(A) × P(B|A). Disjoint events cannot be independent as the occurrence of one rules out the other. Conditional probabilities are best understood
Conditional probability is the probability of an event occurring given that another event has occurred. It is calculated as the probability of both events occurring divided by the probability of the first event. An example is given of calculating the probability of drawing two white balls in succession from an urn without replacement. The formula for conditional probability is derived as the probability of events A and B occurring divided by the probability of A. This is demonstrated using an example of finding the percentage of friends who like chocolate that also like strawberry.
These slides represent a brief idea about conditional probability along with illustrative examples and discussions. It also consists the use of sets to develop a better understanding for the students having the following theorem in their course.
The document discusses smoking-related deaths in the United States each year, including 123,800 from lung cancer out of a total of 438,000 smoking-related deaths. It then defines conditional probability as the probability of one event occurring given that another event has already occurred. Using this definition, it calculates the conditional probability that a smoking-related death will be caused by lung cancer as 28%, based on the number of lung cancer deaths divided by the total number of smoking-related deaths.
This document discusses key concepts in probability. It defines basic terms like experiment, sample space, event, and probability. It provides examples of calculating probability for coin tosses and dice rolls using the classical method of dividing the number of ways an event can occur by the total number of possible outcomes. The document also discusses limitations of the classical method and introduces the empirical and subjective methods of determining probability based on observed frequencies and personal judgment respectively.
Discrete and continuous random variables can be used in various engineering applications. Discrete random variables take on countable values and are used when things are counted, like the number of defective items in a batch. Continuous random variables can take any real number value and are used when measurements are made, like the time for a chemical reaction. Some examples given include using discrete variables to find beam loading at points or quality control sampling, and continuous variables to estimate construction time, structural load magnitude, electrical current amounts, and component failure times.
1. Mathematical expectation, or expected value, of a random variable is calculated by taking the sum of each possible outcome value multiplied by its probability of occurring.
2. For a game with multiple outcomes and their associated probabilities and payoffs, the expected value is the average amount a player can expect to win or lose over many plays. A positive expected value means the game favors the player, while a negative value means it is biased against the player.
3. Variance measures the average amount that values of a random variable deviate from the mean or expected value. Unfortunately, the document does not provide the formulas for variance or standard deviation of a random variable.
This document provides an overview of probability concepts including:
- The three axioms of probability: probabilities are between 0 and 1, the probability of the sample space is 1, and the probability of the union of disjoint events equals the sum of the individual probabilities.
- Formulas for probability, conditional probability, independence, and complements.
- Discrete and continuous random variables and their properties including expected value and variance.
- Examples of probability mass functions for binomial and Poisson distributions.
This document provides an introduction to combinatorics including the sum rule, product rule, permutations, and combinations. The sum rule states that if tasks are independent, the number of ways to do either task is the sum of the number of ways to do each individually. The product rule states that if tasks are independent, the number of ways to do both tasks is the product of the number of ways to do each. Permutations refer to ordered arrangements and combinations refer to unordered arrangements. The document includes examples of applying these concepts.
1. The document discusses basic concepts in probability and statistics, including sample spaces, events, probability distributions, and random variables.
2. Key concepts are explained such as independent and conditional probability, Bayes' theorem, and common probability distributions like the uniform and normal distributions.
3. Statistical analysis methods are introduced including how to estimate the mean and variance from samples from a distribution.
This document discusses the concept of probability. It defines probability as a measure of how likely an event is to occur. Probabilities can be described using terms like certain, likely, unlikely, and impossible. Mathematically, probabilities are often expressed as fractions, with the numerator representing the number of possible outcomes for an event and the denominator representing the total number of possible outcomes. The document provides examples to illustrate concepts like independent and conditional probabilities, as well as complementary events and the gambler's fallacy.
This document discusses sequences and series. It provides definitions of key terms like sequence, finite sequence, infinite sequence, convergent sequence, divergent sequence, monotonic sequence, and geometric progression. It then goes on to solve 4 example problems:
1) It shows that the sequence 2n^2+n/n^2+1 is convergent by taking the limit as n approaches infinity.
2) It uses the ratio test to show that the sequence n!/n^n is convergent.
3) It proves that the sequence 1/1! + 1/2! +...+ 1/n! is convergent by showing it is increasing and bounded.
4) It shows that the sequence
Gaussian elimination is a method for solving systems of linear equations. It involves converting the augmented matrix into an upper triangular matrix using elementary row operations. There are three types of Gaussian elimination: simple elimination without pivoting, partial pivoting, and total pivoting. Partial pivoting interchanges rows to choose larger pivots, while total pivoting searches the whole matrix for the largest number to use as the pivot. Pivoting strategies help prevent zero pivots and reduce round-off errors.
Boolean algebra is an algebra of logic developed by George Boole between 1815-1864 to represent logical statements as an algebra of true and false. It is used to perform logical operations in digital computers by representing true as 1 and false as 0. The fundamental logical operators are AND, OR, and NOT. Boolean algebra expressions can be represented in sum of products (SOP) form or product of sums (POS) form and minimized using algebraic rules or Karnaugh maps. Minterms and maxterms are used to derive Boolean functions from truth tables in canonical SOP or POS form.
This document outlines probability density functions (PDFs) including:
- The definition of a PDF as describing the relative likelihood of a random variable taking a value.
- Properties of PDFs such as being nonnegative and integrating to 1.
- Joint PDFs describing the probability of multiple random variables taking values simultaneously.
- Marginal PDFs describing probabilities of single variables without reference to others.
- An example calculating a joint PDF and its marginals.
This document discusses basic concepts of probability, including:
- The addition rule and multiplication rule for calculating probabilities of compound events.
- Events can be disjoint (mutually exclusive) or not disjoint.
- The probability of an event occurring or its complement must equal 1.
- How to calculate the probability of at least one occurrence of an event using the complement.
- When applying the multiplication rule, you must consider whether events are independent or dependent.
The document discusses diagonalization of matrices. Diagonalization involves finding an invertible matrix P such that P-1AP is a diagonal matrix. The procedure for diagonalizing a matrix A includes: (1) finding the eigenvalues of A, (2) finding linearly independent eigenvectors corresponding to the eigenvalues, (3) forming the matrix P with the eigenvectors as columns, and (4) showing that P-1AP is a diagonal matrix with the eigenvalues along the diagonal. An example demonstrates finding the eigenvectors of a 2x2 matrix A and constructing the matrix P to diagonalize A.
In this presentation we will learn Del operator, Gradient of scalar function , Directional Derivative, Divergence of vector function, Curl of a vector function and after that solved some example related to above.
Gradient in math
Directional derivative in math
Divergence in math
Curl in math
Gradient , Directional Derivative , Divergence , Curl in mathematics
Gradient , Directional Derivative , Divergence , Curl in math
Gradient , Directional Derivative , Divergence , Curl
The document discusses different types of relations, including reflexive, symmetric, transitive, and equivalence relations. It provides examples of each type of relation and defines their key properties. Inverse relations are also discussed, where the ordered pairs of a relation R are reversed to form the inverse relation R-1. An example demonstrates finding the domain, range, and inverse of a given relation defined by an equation.
This document discusses eigen values, eigen vectors, and diagonalization of matrices. It defines eigen values as the roots of the characteristic equation of a matrix. Eigen vectors are non-zero vectors that satisfy AX=λX, where λ is the eigen value. Diagonalization is the process of transforming a matrix A into a diagonal matrix D using a similarity transformation with an invertible matrix P, such that D=P-1AP. The document provides examples to illustrate these concepts and lists various properties of eigen values and eigen vectors.
This document discusses the history and modern applications of probability. It begins by explaining how probability originated from the study of games of chance in the 16th-17th centuries. Mathematicians like Blaise Pascal and Pierre de Fermat helped develop probability theory to aid gamblers. Today, probability is used across many fields like traffic control, genetics, particle physics, and predicting investment returns. The document also defines key probability concepts like outcomes, events, favorable cases, and relative frequency. It provides examples of calculating probability and the formula used.
The document discusses partial differential equations (PDEs). It defines PDEs and gives their general form involving independent variables, dependent variables, and partial derivatives. It describes methods for obtaining the complete integral, particular solution, singular solution, and general solution of a PDE. It provides examples of types of PDEs and how to solve them by assuming certain forms for the dependent and independent variables and their partial derivatives.
This document provides an overview of different number systems including decimal, binary, octal, and hexadecimal. It defines each system, their bases, examples of conversions between them, and their uses in computing. Decimal is base 10, binary is base 2, octal is base 8, and hexadecimal is base 16. Conversions between binary, decimal, octal and hexadecimal are demonstrated through examples. Number systems are important in computing because binary is used for electronic circuitry, octal and hexadecimal allow more compact notation than binary, and decimal is used for basic calculations.
This document explains the Pigeon-Hole Principle and provides an example problem. The Pigeon-Hole Principle states that if the number of items (pigeons) is greater than the number of categories (holes) they are placed into, then at least one category must contain more than one item. The example problem shows 25 students receiving grades of A, B, or C, demonstrating that with more students than grades, at least nine students must have received the same grade. The document is intended to teach this mathematical concept to students through a visual example.
The document discusses the axioms of probability and some basic properties. It defines three axioms for assigning probability values to events in a finite sample space: 1) a probability is between 0 and 1, 2) the probability of the entire sample space is 1, and 3) for mutually exclusive events, the total probability is the sum of the individual probabilities. It then modifies the third axiom for infinite sample spaces and lists five elementary properties that can be obtained from the axioms, such as the probability of the complement of an event being 1 minus the original probability. The document also gives an example problem calculating the probability of not getting a white ball from a bag.
The document discusses key concepts in probability, including:
1) Random phenomena involve outcomes that are unknown but have possible values. Trials produce outcomes that make up events within a sample space.
2) The Law of Large Numbers states that independent repeated events will have a relative frequency that approaches a single probability value.
3) Theoretical probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes, assuming all outcomes are equally likely.
Discrete probability distribution (complete)ISYousafzai
This document discusses discrete random variables. It begins by defining a random variable as a function that assigns a numerical value to each outcome of an experiment. There are two types of random variables: discrete and continuous. Discrete random variables have a countable set of possible values, while continuous variables can take any value within a range. Examples of discrete variables include the number of heads in a coin flip and the total value of dice. The document then discusses how to describe the probabilities associated with discrete random variables using lists, histograms, and probability mass functions.
This document introduces probability and discusses different approaches to defining it. It notes that probability is used to describe variability and uncertainty when outcomes are not certain. Three common definitions of probability are discussed - classical, relative frequency, and subjective - along with their limitations. The document advocates treating probability as a mathematical system defined by axioms rather than worrying about numerical values until a specific application. It then outlines how to construct probability models using sample spaces and assigning probabilities to events based on their composition of simple events.
Random Walks in Statistical Theory of CommunicationIRJET Journal
1. The document discusses random walks, which are a type of random process that can model phenomena like diffusion and stock price variations. Random walks involve successive random steps and can take place in discrete or continuous time and space.
2. The document provides details on modeling random walks mathematically using probabilities and binomial distributions. It also discusses calculating the probability of a random walk returning to its origin.
3. The document shows examples of using random walks to model particle motion in 2D and 3D spaces. It also discusses how continuous random variables can produce Brownian motion, a type of random walk. Random walks have various applications in fields like computer science, image processing, genetics and neuroscience.
This document provides an introduction to combinatorics including the sum rule, product rule, permutations, and combinations. The sum rule states that if tasks are independent, the number of ways to do either task is the sum of the number of ways to do each individually. The product rule states that if tasks are independent, the number of ways to do both tasks is the product of the number of ways to do each. Permutations refer to ordered arrangements and combinations refer to unordered arrangements. The document includes examples of applying these concepts.
1. The document discusses basic concepts in probability and statistics, including sample spaces, events, probability distributions, and random variables.
2. Key concepts are explained such as independent and conditional probability, Bayes' theorem, and common probability distributions like the uniform and normal distributions.
3. Statistical analysis methods are introduced including how to estimate the mean and variance from samples from a distribution.
This document discusses the concept of probability. It defines probability as a measure of how likely an event is to occur. Probabilities can be described using terms like certain, likely, unlikely, and impossible. Mathematically, probabilities are often expressed as fractions, with the numerator representing the number of possible outcomes for an event and the denominator representing the total number of possible outcomes. The document provides examples to illustrate concepts like independent and conditional probabilities, as well as complementary events and the gambler's fallacy.
This document discusses sequences and series. It provides definitions of key terms like sequence, finite sequence, infinite sequence, convergent sequence, divergent sequence, monotonic sequence, and geometric progression. It then goes on to solve 4 example problems:
1) It shows that the sequence 2n^2+n/n^2+1 is convergent by taking the limit as n approaches infinity.
2) It uses the ratio test to show that the sequence n!/n^n is convergent.
3) It proves that the sequence 1/1! + 1/2! +...+ 1/n! is convergent by showing it is increasing and bounded.
4) It shows that the sequence
Gaussian elimination is a method for solving systems of linear equations. It involves converting the augmented matrix into an upper triangular matrix using elementary row operations. There are three types of Gaussian elimination: simple elimination without pivoting, partial pivoting, and total pivoting. Partial pivoting interchanges rows to choose larger pivots, while total pivoting searches the whole matrix for the largest number to use as the pivot. Pivoting strategies help prevent zero pivots and reduce round-off errors.
Boolean algebra is an algebra of logic developed by George Boole between 1815-1864 to represent logical statements as an algebra of true and false. It is used to perform logical operations in digital computers by representing true as 1 and false as 0. The fundamental logical operators are AND, OR, and NOT. Boolean algebra expressions can be represented in sum of products (SOP) form or product of sums (POS) form and minimized using algebraic rules or Karnaugh maps. Minterms and maxterms are used to derive Boolean functions from truth tables in canonical SOP or POS form.
This document outlines probability density functions (PDFs) including:
- The definition of a PDF as describing the relative likelihood of a random variable taking a value.
- Properties of PDFs such as being nonnegative and integrating to 1.
- Joint PDFs describing the probability of multiple random variables taking values simultaneously.
- Marginal PDFs describing probabilities of single variables without reference to others.
- An example calculating a joint PDF and its marginals.
This document discusses basic concepts of probability, including:
- The addition rule and multiplication rule for calculating probabilities of compound events.
- Events can be disjoint (mutually exclusive) or not disjoint.
- The probability of an event occurring or its complement must equal 1.
- How to calculate the probability of at least one occurrence of an event using the complement.
- When applying the multiplication rule, you must consider whether events are independent or dependent.
The document discusses diagonalization of matrices. Diagonalization involves finding an invertible matrix P such that P-1AP is a diagonal matrix. The procedure for diagonalizing a matrix A includes: (1) finding the eigenvalues of A, (2) finding linearly independent eigenvectors corresponding to the eigenvalues, (3) forming the matrix P with the eigenvectors as columns, and (4) showing that P-1AP is a diagonal matrix with the eigenvalues along the diagonal. An example demonstrates finding the eigenvectors of a 2x2 matrix A and constructing the matrix P to diagonalize A.
In this presentation we will learn Del operator, Gradient of scalar function , Directional Derivative, Divergence of vector function, Curl of a vector function and after that solved some example related to above.
Gradient in math
Directional derivative in math
Divergence in math
Curl in math
Gradient , Directional Derivative , Divergence , Curl in mathematics
Gradient , Directional Derivative , Divergence , Curl in math
Gradient , Directional Derivative , Divergence , Curl
The document discusses different types of relations, including reflexive, symmetric, transitive, and equivalence relations. It provides examples of each type of relation and defines their key properties. Inverse relations are also discussed, where the ordered pairs of a relation R are reversed to form the inverse relation R-1. An example demonstrates finding the domain, range, and inverse of a given relation defined by an equation.
This document discusses eigen values, eigen vectors, and diagonalization of matrices. It defines eigen values as the roots of the characteristic equation of a matrix. Eigen vectors are non-zero vectors that satisfy AX=λX, where λ is the eigen value. Diagonalization is the process of transforming a matrix A into a diagonal matrix D using a similarity transformation with an invertible matrix P, such that D=P-1AP. The document provides examples to illustrate these concepts and lists various properties of eigen values and eigen vectors.
This document discusses the history and modern applications of probability. It begins by explaining how probability originated from the study of games of chance in the 16th-17th centuries. Mathematicians like Blaise Pascal and Pierre de Fermat helped develop probability theory to aid gamblers. Today, probability is used across many fields like traffic control, genetics, particle physics, and predicting investment returns. The document also defines key probability concepts like outcomes, events, favorable cases, and relative frequency. It provides examples of calculating probability and the formula used.
The document discusses partial differential equations (PDEs). It defines PDEs and gives their general form involving independent variables, dependent variables, and partial derivatives. It describes methods for obtaining the complete integral, particular solution, singular solution, and general solution of a PDE. It provides examples of types of PDEs and how to solve them by assuming certain forms for the dependent and independent variables and their partial derivatives.
This document provides an overview of different number systems including decimal, binary, octal, and hexadecimal. It defines each system, their bases, examples of conversions between them, and their uses in computing. Decimal is base 10, binary is base 2, octal is base 8, and hexadecimal is base 16. Conversions between binary, decimal, octal and hexadecimal are demonstrated through examples. Number systems are important in computing because binary is used for electronic circuitry, octal and hexadecimal allow more compact notation than binary, and decimal is used for basic calculations.
This document explains the Pigeon-Hole Principle and provides an example problem. The Pigeon-Hole Principle states that if the number of items (pigeons) is greater than the number of categories (holes) they are placed into, then at least one category must contain more than one item. The example problem shows 25 students receiving grades of A, B, or C, demonstrating that with more students than grades, at least nine students must have received the same grade. The document is intended to teach this mathematical concept to students through a visual example.
The document discusses the axioms of probability and some basic properties. It defines three axioms for assigning probability values to events in a finite sample space: 1) a probability is between 0 and 1, 2) the probability of the entire sample space is 1, and 3) for mutually exclusive events, the total probability is the sum of the individual probabilities. It then modifies the third axiom for infinite sample spaces and lists five elementary properties that can be obtained from the axioms, such as the probability of the complement of an event being 1 minus the original probability. The document also gives an example problem calculating the probability of not getting a white ball from a bag.
The document discusses key concepts in probability, including:
1) Random phenomena involve outcomes that are unknown but have possible values. Trials produce outcomes that make up events within a sample space.
2) The Law of Large Numbers states that independent repeated events will have a relative frequency that approaches a single probability value.
3) Theoretical probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes, assuming all outcomes are equally likely.
Discrete probability distribution (complete)ISYousafzai
This document discusses discrete random variables. It begins by defining a random variable as a function that assigns a numerical value to each outcome of an experiment. There are two types of random variables: discrete and continuous. Discrete random variables have a countable set of possible values, while continuous variables can take any value within a range. Examples of discrete variables include the number of heads in a coin flip and the total value of dice. The document then discusses how to describe the probabilities associated with discrete random variables using lists, histograms, and probability mass functions.
This document introduces probability and discusses different approaches to defining it. It notes that probability is used to describe variability and uncertainty when outcomes are not certain. Three common definitions of probability are discussed - classical, relative frequency, and subjective - along with their limitations. The document advocates treating probability as a mathematical system defined by axioms rather than worrying about numerical values until a specific application. It then outlines how to construct probability models using sample spaces and assigning probabilities to events based on their composition of simple events.
Random Walks in Statistical Theory of CommunicationIRJET Journal
1. The document discusses random walks, which are a type of random process that can model phenomena like diffusion and stock price variations. Random walks involve successive random steps and can take place in discrete or continuous time and space.
2. The document provides details on modeling random walks mathematically using probabilities and binomial distributions. It also discusses calculating the probability of a random walk returning to its origin.
3. The document shows examples of using random walks to model particle motion in 2D and 3D spaces. It also discusses how continuous random variables can produce Brownian motion, a type of random walk. Random walks have various applications in fields like computer science, image processing, genetics and neuroscience.
This document provides an introduction to probability and important concepts in probability theory. It defines probability as a measure of the likelihood of an event occurring based on chance. Probability can be estimated empirically by calculating the relative frequency of outcomes in a series of trials, or estimated subjectively based on experience. Classical probability uses an a priori approach to assign probabilities to outcomes that are considered equally likely, such as outcomes of rolling dice or drawing cards. The document provides examples and definitions of key probability terms and concepts such as sample space, events, axioms of probability, and approaches to calculating probability.
The document discusses probability trees and their use in calculating probabilities of events. It provides an example using probability trees to calculate the probability that a flight leaving Srinagar belonged to a particular airline, given that the flight was on time. It is stated that probability trees allow visualizing randomness and conditional probabilities in a simple way and can be used to calculate the probabilities of multiple simultaneous events. The document concludes by stating that probability trees allow comparing the outcomes of different decisions and speed up probability calculations.
The document provides an overview of topics in business mathematics including linear equations, cost-output functions, matrices, logarithmic and exponential functions, and their applications in business and real life. Examples of applications given include using linear equations to model costs, revenues, and profits; break-even analysis; modeling population growth with exponential functions; and using derivatives and integrals to solve problems in physics, engineering, and economics. The document aims to demonstrate how mathematics is relevant across many domains including business, science, and daily life.
This document summarizes recent research on trajectory planning algorithms for autonomous vehicles. It discusses graph search algorithms like A* that plan optimal paths but have limitations in dynamic environments. Improvements like D* and Focused D* allow recomputing only changed portions of the path. Kinematic A* adds vehicle constraints to generate smoother, safer paths. Overall, the document analyzes how these algorithms aim to enable reliable trajectory planning in unknown, changing environments.
This document discusses different types of simulation models. It begins by introducing deterministic simulation models, where events and variable relationships are governed by known rules. It then discusses stochastic simulation models which incorporate random variables. The document also distinguishes between static models, which do not involve time, and dynamic models which simulate a system over time. It provides examples of deterministic simulations, such as simulating a loan repayment, and stochastic simulations, such as estimating pi using Monte Carlo methods.
Probability is the branch of mathematics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur.[note 1][1][2] The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ('heads' and 'tails') are both equally probable; the probability of 'heads' equals the probability of 'tails'; and since no other outcomes are possible, the probability of either 'heads' or 'tails' is 1/2 (which could also be written as 0.5 or 50%).
These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in areas of study such as statistics, mathematics, science, finance, gambling, artificial intelligence, machine learning, computer science, game theory, and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.
This document provides an introduction to probability. It defines probability as a measure of how likely an event is to occur. Probability is expressed as a ratio of favorable outcomes to total possible outcomes. The key terms used in probability are defined, including event, outcome, sample space, and elementary events. The theoretical approach to probability is discussed, where probability is predicted without performing the experiment. Random experiments are described as those that may not produce the same outcome each time. Laws of probability are presented, such as a probability being between 0 and 1. Applications of probability in everyday life are mentioned, such as reliability testing of products. Two example probability problems are worked out.
The document defines key terms related to probability distributions, including probability distribution, random variable, discrete and continuous distributions. It describes different probability distributions - binomial, hypergeometric, and Poisson - and how to calculate probabilities, means, variances, and standard deviations for each. Examples are provided to illustrate concepts like computing probabilities using the binomial distribution for coin tosses or late flights, and the hypergeometric distribution for selecting employees for a committee.
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This research paper demonstrates the invention of the kinetic bands, based on Romanian mathematician and statistician Octav Onicescu’s kinetic energy, also known as “informational energy”, where we use historical data of foreign exchange currencies or indexes to predict the trend displayed by a stock or an index and whether it will go up or down in the future. Here, we explore the imperfections of the Bollinger Bands to determine a more sophisticated triplet of indicators that predict the future movement of prices in the Stock Market. An Extreme Gradient Boosting Modelling was conducted in Python using historical data set from Kaggle, the historical data set spanning all current 500 companies listed. An invariable importance feature was plotted. The results displayed that Kinetic Bands, derived from (KE) are very influential as features or technical indicators of stock market trends. Furthermore, experiments done through this invention provide tangible evidence of the empirical aspects of it. The machine learning code has low chances of error if all the proper procedures and coding are in play. The experiment samples are attached to this study for future references or scrutiny.
This document discusses probability and its key concepts. It begins by defining probability as a quantitative measure of uncertainty ranging from 0 to 1. Probability can be understood objectively based on problems or subjectively based on beliefs. Key probability concepts discussed include:
- Sample space, simple events, and compound events
- Classical, relative frequency, and subjective approaches to assigning probabilities
- Complement, intersection, and union of events
- Conditional probability and independence of events
- Rules for calculating probabilities of combined events like the multiplication rule
Examples are provided to illustrate concepts like defining sample spaces, calculating probabilities of individual and combined events, determining conditional probabilities, and assessing independence. Overall, the document provides a comprehensive overview of fundamental probability
Intro to Quant Trading Strategies (Lecture 10 of 10)Adrian Aley
This document provides an overview of risk management strategies for algorithmic trading. It discusses various risk measurement techniques including Value at Risk (VaR), Extreme Value Theory (EVT), and the generalized extreme value distribution. Specific risks for different asset classes like bonds, stocks, derivatives and currencies are outlined. Monte Carlo simulation is presented as a technique for modeling rare events and fat tails in return distributions. The document emphasizes that risk is multifaceted and not fully captured by any single measure.
The document provides an introduction to probability theory, including definitions of key terms like trial, event, exhaustive events, favorable events, independent events, mutually exclusive events, and equally likely events. It discusses three approaches to defining probability: classical, statistical, and axiomatic. The classical approach defines probability as the ratio of favorable cases to total possible cases. The statistical approach determines probabilities based on empirical observations over many trials. The axiomatic approach uses set theory and axioms to define probability without restrictions of previous approaches.
A new Evolutionary Reinforcement Scheme for Stochastic Learning Automatainfopapers
F. Stoica, E. M. Popa, A new Evolutionary Reinforcement Scheme for Stochastic Learning Automata, Proceedings of the 12th WSEAS International Conference on COMPUTERS, Heraklion, Greece, July 23-25, ISBN: 978-960-6766-85-5, ISSN: 1790-5109, pp. 268-273, 2008
International Journal of Engineering Inventions (IJEI) provides a multidisciplinary passage for researchers, managers, professionals, practitioners and students around the globe to publish high quality, peer-reviewed articles on all theoretical and empirical aspects of Engineering and Science.
This document provides an overview of basic probability theory concepts including probability, mutually exclusive events, and independence. It discusses probability as a measure of likelihood between 0 and 1. Key concepts covered include interpretations of probability, the mathematical treatment including independent, conditional, and summary probabilities, and applications in areas like reliability and natural language processing. Mutually exclusive events are defined as events that cannot occur simultaneously, while independent events have probabilities that are unaffected by each other.
Differential game theory for Traffic Flow ModellingSerge Hoogendoorn
Lecture given at the INdAM symposium in Rome, 2017. The lecture shows how you can use differential games to model traffic flows, focussing on pedestrian simulation.
Similar to A report on application of probability to control the flow of traffic through a highway system (20)
This document discusses Java packages. It defines a package as a way to group related classes, interfaces, and sub-packages. There are two types of packages - predefined packages that come with Java, and user-defined packages created by the programmer. Packages can be created using the package keyword and accessed using import statements. Packages provide advantages like reusability, security, and preventing naming conflicts, but also have disadvantages like not being able to pass parameters to packages.
The Raspberry Pi is a small, low-cost computer that can connect to a keyboard, mouse and monitor. It runs Linux and allows users to explore programming through languages like Scratch and Python. The Raspberry Pi has GPIO pins that enable controlling electronic components for physical computing projects. It comes in different models that vary in processing power, memory and connectivity options. Python's RPi.GPIO module allows controlling the GPIO pins to take sensor input and drive outputs. Code examples demonstrate setting pin modes, reading/writing pin values and using an IR sensor for obstacle avoidance.
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laplace transform of function of the 풕^풏f(t)ABHIJITPATRA23
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(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 2)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐈𝐂𝐓 𝐢𝐧 𝐞𝐝𝐮𝐜𝐚𝐭𝐢𝐨𝐧:
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𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐫𝐞𝐥𝐢𝐚𝐛𝐥𝐞 𝐬𝐨𝐮𝐫𝐜𝐞𝐬 𝐨𝐧 𝐭𝐡𝐞 𝐢𝐧𝐭𝐞𝐫𝐧𝐞𝐭:
-Students will be able to discuss what constitutes reliable sources on the internet. They will learn to identify key characteristics of trustworthy information, such as credibility, accuracy, and authority. By examining different types of online sources, students will develop skills to evaluate the reliability of websites and content, ensuring they can distinguish between reputable information and misinformation.
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2. TOPIC- A REPORT ON APPLICATION OF PROBABILITY TO CONTROLTHE
FLOWOF TRAFFIC THROUGH A HIGHWAY SYSTEM
PROJECT- 1
NAME- ABHIJIT PATRA
3. CONTENT
PROBABILITY AND ITS TERMINOLOGY
CALCULATION
EXAMPLE
APPLICATION OF PROBABILITY TO
CONTROL TRAFFIC FLOW
4. PROBABILITYANDRELATEDTERMINOLOGY
Probability means the mathematical chance that something might happen, is used in a numerous
day-to-day applications, including in weather forecasts, sports strategies, insurance options, games
and recreational activities, making business.
In other word probability is the branch of mathematics concerning numerical descriptions of how
likely an event is to occur or how like the proposition is true.
Probability of an event is a number between 0 and 1 where 0 indicates impossible event and 1
indicates certain events.
Event: – an event is a set of outcomes of an experiment to which probability is assigned. Event is
the subset of sample space
Sample space: – sample space of an experiment is the set of all possible outcomes or results of that
experiment. A sample space is usually denoted as a set notation and every outcomes are listed as
elements of the set.
5. Types of events
1. Mutually exclusive events:
The events are said to be mutually exclusive or disjoint when they can not occur at the same
time
Example- rolling a die once. The outcome a) odd number
b) even number
Here the two events a and b are mutually exclusive event.
2. Independent event:
The event are said to be independent if the occurrence of one does not affect the probability
of occurrence of other .
Example- if we flip a coin in the air and get the outcome as head ,then again if we flip
the coin but this time we get the outcomes as tail. In both cases the occurrence of both the events
is independent of each other.
6. CALCULATION
Probability of an event =
sample space = S
Event occurred = E
Probability of a is denoted as P(E)
P(E) =
NUMBER OF GIVEN OUTCOME
NUMBER OF ALL POSSIBLE OUTCOME
|E|
|S|
7. EXAMPLE
Let consider two dice rolled once . The event is sum of the outcome is even.
Find the probability of the event.
Ans- If two dice rolled once the sample space s=
{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),
(2,6),(3,1),(3,2),(3,3),(3,4),(3,5), (3,6),(4,1),(4,2),(4,3),(4,4)
(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3)
,(6,4)(6,5)(6,6)}
|S|=𝟔𝟐
Event E= the sum of the outcome is even = {(1,1),(1,3),(1,5),(2,2),(2,4),(2,6),(3,1),(3,3),(3,5),
(4,2),(4,4),(4,6) (5,1),(5,3),(5,5),(6,2),(6,4),(6,6)}
|E|=18
P(E) =
|𝑬|
|𝑺|
=
18
36
8. APPLICATIONOF PROBABILITY TO CONTROL THE TRAFFIC FLOW
The aim of this model is to provide a simple yet accurate representation of lane changing.
We observe the probability a specific car will switch at a given time, which is accurate from
the macroscopic point of view.
This approach is reasonable because there is no definite rule to account for drivers’
decisions to switch lanes: even if switching lanes is objectively correct in a given situation,
the driver may not make the best decisions.
Our model follows the ‘keep right except to pass’ (KREP) rule, widely used in traffic
systems around the world. Cars stay in the rightmost lane unless the car in front of them is
driving below their preferred speed.
9. In that case, they could switch to the next lane on the left until they are able to drive at their preferred speed,
and continue driving on that lane until they overtake the slower car and have enough empty distance to
return to the right lane
If a car in any lane (except the leftmost) is traveling at a lower speed than its desired speed because it is
impeded by the car in front, then there is a chance the driver will decide to switch to the left lane.
First, however, the driver must take into account safety concerns — it must be separated enough from
neighboring cars to be able to switch lanes safely.
The “two-second rule”, a common practice among drivers for car-following, forbids changing lanes if a car
is less than two seconds behind the car in front of it or behind it.
CONTINUED….
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