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# Exam 2 - Chapters 4,5,&6 Probability (Continuous & Discrete) OPRE 3360

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This 8 page Study Guide was uploaded by Vivek Kotikalapudi on Saturday October 15, 2016. The Study Guide belongs to OPRE 3360 at University of Texas at Dallas taught by Ilhan Emre Ertan in Fall 2016. Since its upload, it has received 59 views. For similar materials see Managerial Methods in Decision Making Under Uncertainty in Business at University of Texas at Dallas.

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Date Created: 10/15/16

Managerial Methods in Decision-Making Under Uncertainty, Exam 2 Study Guide Instructor: Ilhan Emre Ertan Chapters 4,5,6 Chapter 4: Introduction to Probability Managers often base their decision on analysis of uncertainties. 1. What is probability? - A numerical measure of the likelihood that an event will occur. - Probability values are always assigned on a scale from 0 to 1. - A probability near zero indicates that an event is quite unlikely to occur while a probability near one one indicates an event is almost certain to occur. Definitions: Experiment: Any process that generates well-defined outcomes. Sample Space: The set of all possible experimental outcomes. An experimental outcome is also called a sample point. Probability determines the outcomes: Even if a statistical experiment is repeated in exactly the same way, an entirely different outcome may occur. Also known as random experiments. On any single repetition of an experiment, one and only one of the possible experimental outcomes will occur. Examples: Experiment Experimental Outcomes Tossing a coin Head; Tail Inspection of a part Defective; Non- defective Conduct a sales call Purchase; No purchase Rolling a die 1;2;3;4;5;6 Playing a soccer Win; Lose; Tie game The probability assigned to each experimental outcome must be between 0 and 1, inclusively, 0 ≤ P(E) ≤ 1 for all i The sum of the probabilities for all experimental outcomes must equal 1, where n is the number of experimental outcomes. Methods for Assigning Probabilities: Classical Method – Assigning probabilities based on the assumption of equally likely outcomes. Relative Frequency Method – Assigning probabilities based on experimentation or historical data. Subjective Method – Assigning probabilities based on judgment. Classical Method: If an experiment has n possible outcomes, the classical method would assign a probability of 1/n to each outcome. Example: Experiment: Rolling a Die Sample Space: S = {1,2,3,4,5,6} Probabilities: Each sample point has a 1/6 chance of occurring An important component of the classical method is counting the number of possible outcomes. Counting Rule #1: If an experiment consists of a sequence of k steps in which there are n p1ssible results for the first step, n 2ossible results for the second step, and so on. - - - The total number of outcomes is given by (n )(1 ) 2 . . (nk) - To keep it simple, each outcome is multiplied by the next outcome of the next step. - Good graphical representation = Tree diagram - A tree diagram is a multiple-step diagram that allows us to calculate the probability of an outcome by multiplying along the branches. Example: Counting Rule #2: Combinations: N objects taken n at a time (order of n objects is not important) - The number of experimental outcomes when n objects are to be selected from a set of N objects. where: N! = N(N-1)(N-2)…(2)(1) n! = n(n-1)(n-2)…(2)(1) 0! = 1 N N! C n Counting Rule #3: Permutations: N objects taken n at a time (order of n objects IS important) - The number of experimental outcomes when n objects are to be selected from a set of N objects where the order of selection is important. Overall: Classical Method Summary: - The classical method measures the likelihood of something happening, but also, contains elements that have an equal likelihood of occurring in a countable way. - The total number of outcomes can be measured by multiplying each outcome along the way. - Combinations shows the number of experimental outcomes taken from a set of objects where order does not matter. - In Permutations, the order of the set DOES matter. Relative Frequency Method: - Each probability assignment is given by dividing the frequency by the total frequency. Subjective Method: - Used when economic conditions or a company’s circumstances change rapidly. - A probability value should express our degree of belief that the experimental outcome will occur. - The best probability estimates often are obtained by combining the estimates from the classical or relative frequency approach with the subjective estimate. Event: - A collection of sample points. - The probability of any event is equal to the sum of the probabilities of the sample points in the event. Basic Relationships of Probabilities - Complement of an event - The complement of event A is defined as the event consisting of all sample points that are not in A. AB - Union of two events - The union of events A and B is the event containing all sample points that are in A or B or both. AB - Intersection of two events - The intersection of events A and B is the set of all sample points that are in both A and B. - Addition Law - The addition law provides a way to compute the probability of event A, or B, or both A and B occurriP AB P A P B P AB - Mutually exclusive events - Two events are said to be mutually exclusive if the events have no sample points in common. - If events A and B are mutually exclusive: PAB 0 - The addition law for mutually exclusive events is: P AB P A P B - Conditional Probability - The probability of an event given another event has occurred is called a conditional probability. P A| B P AB P B - Multiplication Law - The multiplication law provides a way to compute the probability of the intersection of two events. - Can also be used as a test to see if two events are independent. P AB P B P A|B - Independent Events - If the probability of event A is not changed by the existence of event B, we say that events A and B are independent. - Two events A and B are independent PA| B PA o r PB| A PB Mutual Exclusiveness and Independence: - Two events with nonzero probabilities cannot be both mutually exclusive and independent. - If one mutually exclusive event is known to occur, the other cannot occur; thus, the probability of the other event occurring is reduced to zero (and they are therefore dependent) - Two events that are not mutually exclusive, might or might not be independent. Chapters 5 & 6 Discrete and Continuous probability distributions Random Variable – A numerical description of the outcome of an experiment. Discrete random variable - may assume either a finite number of values or an infinite sequence of values. Continuous random variable - may assume any numerical value in an interval or collection of intervals. The discrete probability distribution for a discrete random variable describes how probabilities are distributed over the point values of the random variable The continuous probability distribution of a continuous random variable assumes a value within some given interval from x to x is 1 2 defined to be the area under the graph of the probability density function between x a1d x 2 Probability of point values are equal to “zero”! The probability distribution is defined by a probability function, denoted by f(x), which provides the probability for each value of the random variable. The required conditions for a discrete probability function are: f x 0 f x 1 The required conditions for a continuous probability function are: f x 0,foa nsp eci if cx f(x) 1 Continuous Probability Distributions: A continuous random variable can assume any value in an interval on the real line or in a collection of intervals It is not possible to talk about the probability of the random variable assuming a particular value P x (any continuous value) = 0 Instead, we talk about the probability of the random variable assuming a value within a given interval The probability of the random variable assuming a value within some given interval from x1to x2is defined to be the area under the graph of the probability density function between x1and x 2 Discrete Uniform Probability Distribution: The discrete uniform probability function: - n = the number of values the random variable may assume. The expected value, or mean, of a random variable is a measure of its central location. The expected value is a weighted average of the values the random variable may assume. The weights are the probabilities. The expected value does not have to be a value the random variable can assume. The variance summarizes the variability in the values of a random variable 2 2 V rax x f x The variance is a weighted average of the squared deviations of a random variable from its mean. The weights are the probabilities. The standard deviation, σ , is defined as the positive square root of the variance. Four Properties of a Binomial Experiment: The experiment consists of a sequence of n identical trials. Two outcomes, success or failure, are possible on each trial. The probability of a success, denoted by p, does not change from trial to trial. The trials are independent. Our interest is in the # of successes occurring in the n trials. Let x = the # of successes occurring in the n trials. Binomial Probability Function: n! nx x = the number of successes f x p 1 p x! nx! p = the probability of a success on one trial n = the number of trials f(x) = the probability of x successes in n trials n! = n(n-1)(n-2)…(2)(1) Expected Value: Ex n p 2 Variance: V rax n 1 p Standard Deviation: n p1 p Continuous Uniform Probability Distribution: A random variable is uniformly distributed whenever the probability is proportional to the interval’s length The uniform probability density function is: f (x) = 1/(b – a) for a < x < b = 0 elsewhere where : a = smallest value the variable can assume b = largest value the variable can assume Expected Value of x: Variance of x: ba2 E x ab Varx 12 2 Area as a Measure of Probability: The area under the graph of f(x) and probability are identical This is valid for all continuous random variables The probability that x takes on a value between some lower value x 1 and some higher value x ca2 be found by computing the area under the graph of f(x) over the interval from x t1 x 2 Normal Probability Distribution: The normal probability distribution is the most important distribution for describing a continuous random variable It is widely used in statistical inference It has been used in a wide variety of applications including Heights of people Rainfall amounts Test scores Scientific measurements Normal Probability Density Function: 1 x 2 2 f x e 2 where: m = mean s = standard deviation p = 3.14159 e = 2.71828 Standard Normal Probability Distribution: A random variable having a normal distribution with mean=0 and standard deviation=1 is said to have a standard normal probability distribution The letter z is used to designate the standard normal random variable Converting to the standard normal distribution: z is the measure of the number of standard deviations x is away from m x z

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