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**Probability Distributions**

Chapter 4 Probability Distributions Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Discrete Random Variables**

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**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

Random Variable For a given sample space S of some experiment, a random variable is any rule that associates a number with each outcome in S . Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Types of Random Variables A discrete random variable is an rv whose possible values either constitute a finite set or else can listed in an infinite sequence. A random variable is continuous if its set of possible values consists of an entire interval on a number line. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

Random Variables Represents a possible numerical value from a random event Random Variables Discrete Random Variable Continuous Random Variable Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Probability Distributions for Discrete Random Variables**

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**Discrete Random Variables**

Can only assume a countable number of values Examples: Roll a die twice Let x be the number of times 4 comes up (then x could be 0, 1, or 2 times) Toss a coin 5 times. Let x be the number of heads (then x = 0, 1, 2, 3, 4, or 5) Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Discrete Probability Distribution**

Experiment: Toss 2 Coins. Let x = # heads. 4 possible outcomes Probability Distribution T T x Value Probability /4 = .25 /4 = .50 /4 = .25 T H H T .50 .25 Probability H H x Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Discrete Probability Distribution**

The set of ordered Pairs (x, f(x) ) is a probability function, probability mass function or probability distribution function of the discrete random variable X if f(x) 0 P(X=x) = f(x) Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

Cumulative Distribution Function The cumulative distribution function (cdf) F(x) of a discrete rv variable X with pmf p(x) is defined for every number by For any number x, F(x) is the probability that the observed value of X will be at most x. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Proposition For any two numbers a and b with “a–” represents the largest possible X value that is strictly less than a. Note: For integers Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Example A probability distribution for a random variable X: x –8 –3 –1 1 4 6 P(X = x) 0.13 0.15 0.17 0.20 0.11 0.09 Find 0.65 0.67 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Expected Values of Discrete Random Variables**

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The Expected Value of X Let X be a discrete rv with possible values x and pmf p(x). The expected value or mean value of X, denoted Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Ex. Use the data below to find out the expected number of the number of credit cards that a student will possess. x = # credit cards x P(x =X) 0.08 1 0.28 2 0.38 3 0.16 4 0.06 5 0.03 6 0.01 =1.97 About 2 credit cards Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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The Expected Value of a Function If the rv X has the set of possible values x and pmf p(x), then the expected value of any function h(x), denoted Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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The Variance and Standard Deviation Let X have pmf p(x), and expected value Then the variance of X, denoted V(X) The standard deviation (SD) of X is Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Ex. The quiz scores for a particular student are given below: 22, 25, 20, 18, 12, 20, 24, 20, 20, 25, 24, 25, 18 Find the variance and standard deviation. Value 12 18 20 22 24 25 Frequency 1 2 4 3 Probability .08 .15 .31 .23 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Shortcut Formula for Variance Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Mean and Variance of Linear Combinations of Random Variables Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Expected Value of Linear Combinations This leads to the following: For any constant a, 2. For any constant b, Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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The Expected Value of Also Finally Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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If X and Y are independent, then Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Rules of Variance and This leads to the following: Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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If X and Y are random variables, then If X and Y are Independent random variables, then Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

If X and Y are Independent random variables, then If X1, X2, … Xn are Independent random variables, then Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Special Distributions for Discrete Random Variables**

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**The Discrete Uniform Distribution**

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If the random variable X assumes the values x1, x2, …, xk with equal probability, then the discrete uniform distribution is given by The mean and the variance of the discrete uniform are respectively Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**The Binomial Probability Distribution**

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Binomial Experiment An experiment for which the following four conditions are satisfied is called a binomial experiment (Bernoulli Process). The experiment consists of a sequence of n trials, where n is fixed in advance of the experiment. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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The trials are identical, and each trial can result in one of the same two possible outcomes, which are denoted by success (S) or failure (F). The trials are independent. The probability of success is constant from trial to trial: denoted by p. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Binomial Random Variable Given a binomial experiment consisting of n trials, the binomial random variable X associated with this experiment is defined as X = the number of S’s among n trials Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Notation for the pmf of a Binomial rv**

Because the pmf of a binomial rv X depends on the two parameters n and p, we denote the pmf by b(x;n,p). Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Computation of a Binomial pmf**

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**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

Ex. A card is drawn from a standard 52-card deck. If drawing a club is considered a success, find the probability of a. exactly one success in 4 draws (with replacement). p = ¼; q = 1– ¼ = ¾ b. no successes in 5 draws (with replacement). Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Notation for cdf For X ~ Bin(n, p), the cdf will be denoted by x = 0, 1, 2, …n Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Mean and Variance For X ~ Bin(n, p), then E(X) = np, and V(X) = np(1 – p) = npq, (where q = 1 – p). Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

Ex. 5 cards are drawn, with replacement, from a standard 52-card deck. If drawing a club is considered a success, find the mean, variance, and standard deviation of X (where X is the number of successes). p = ¼; q = 1– ¼ = ¾ Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Ex. If the probability of a student successfully passing this course (C or better) is 0.82, find the probability that given 8 students a. all 8 pass. b. none pass. c. at least 6 pass. = Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Hypergeometric and Distributions**

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**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

The Hypergeometric Distribution The three assumptions that lead to a hypergeometric distribution: 1. The population or set to be sampled consists of N individuals, objects, or elements (a finite population). Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Each individual can be characterized as a success (S) or failure (F), and there are K successes in the population. A sample of n individuals is selected without replacement in such a way that each subset of size n is equally likely to be chosen. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Hypergeometric Distribution If X is the number of S’s in a completely random sample of size n drawn from a population consisting of K S’s and (N – K) F’s, then the probability distribution of X, called the hypergeometric distribution, is given by Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

Ex. A card is drawn from a standard 52-card deck. If drawing a club is considered a success, find the probability of a. exactly one success in 4 draws (without replacement). N=52, K=13, n=4 b. no successes in 5 draws (without replacement). Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Hypergeometric Mean and Variance**

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Geometric Distribution The pmf of the geometric rv X with parameter p = P(S) is x = 1, 2, … Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Geometric Mean and Variance**

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**The Poisson Probability Distribution**

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Poisson Process Assumptions: 1. The number of outcomes occurring in one time interval or specified region of space is independent of the number that occurs in any other disjoint time interval or specified region. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Poisson Process 2. The probability that a single outcome will occur during a very short time interval or specified region is proportional to the length of time interval or the size of the region and does not depend on the number of outcomes occurring outside interval or specified region 3.The Probability that more than one outcome will occur in such short time interval or fall in such small region is negligible. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Poisson Distribution The probability Distribution of the Poisson random variable X, representing the number of outcomes occurring in a given time interval or specified region denoted by t is where is the average number of outcomes per unit time or region. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**The Poisson Distribution as a Limit of the binomial**

Suppose that in the binomial pmf b(x;n, p), we let in such a way that np approaches a value Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Poisson Distribution Mean and Variance**

If X has a Poisson distribution with parameter Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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Ex. At a department store customers arrive at a checkout counter at a rate on three per 15 minutes. What is the probability that a counter has two customers over a 30 minutes randomly selected? Sol. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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