CHAPTER 5 Discrete Probability Distributions Copyright Mc GrawHill
CHAPTER 5 Discrete Probability Distributions © Copyright Mc. Graw-Hill 2004 1
Objectives n Construct a probability distribution for a random variable. n Find the mean, variance, and expected value for a discrete random variable. n Find the exact probability for X successes in n trials of a binomial experiment. © Copyright Mc. Graw-Hill 2004 2
Objectives (cont’d. ) n Find the mean, variance, and standard deviation for the variable of a binomial distribution. n Find probabilities for outcomes of variables using the Poisson, hypergeometric, and multinomial distributions. © Copyright Mc. Graw-Hill 2004 3
Introduction n Many decisions in business, insurance, and other reallife situations are made by assigning probabilities to all possible outcomes pertaining to the situation and then evaluating the results. © Copyright Mc. Graw-Hill 2004 4
Introduction (cont’d. ) n This chapter explains the concepts and applications of probability distributions. In addition, special probability distributions, such as the binomial, multinomial, Poisson, and hypergeometric distributions are explained. © Copyright Mc. Graw-Hill 2004 5
Random Variables n A random variable is a variable whose values are determined by chance. © Copyright Mc. Graw-Hill 2004 6
Discrete Probability Distribution n A discrete probability distribution consists of the values a random variable can assume and the corresponding probabilities of the values. The probabilities are determined theoretically or by observation. © Copyright Mc. Graw-Hill 2004 7
Calculating the Mean n In order to find the mean for a probability distribution, one must multiply each possible outcome by its corresponding probability and find the sum of the products. © Copyright Mc. Graw-Hill 2004 8
Rounding Rule n The mean, variance, and standard deviation should be rounded to one more decimal place than the outcome, X. © Copyright Mc. Graw-Hill 2004 9
Variance of a Probability Distribution n The variance of a probability distribution is found by multiplying the square of each outcome by its corresponding probability, summing those products, and subtracting the square of the mean. n The formula for calculating the variance is: n The formula for the standard deviation is: © Copyright Mc. Graw-Hill 2004 10
Expected Value n Expected value or expectation is used in various types of games of chance, in insurance, and in other areas, such as decision theory. © Copyright Mc. Graw-Hill 2004 11
Expected Value (cont’d. ) n The expected value of a discrete random variable of a probability distribution is theoretical average of the variable. The formula is: n The symbol E(X) is used for the expected value. © Copyright Mc. Graw-Hill 2004 12
The Binomial Distribution n Many types of probability problems have only two possible outcomes or they can be reduced to two outcomes. n Examples include: when a coin is tossed it can land on heads or tails, when a baby is born it is either a boy or girl, etc. © Copyright Mc. Graw-Hill 2004 13
The Binomial Experiment n The binomial experiment is a probability experiment that satisfies these requirements: 1. Each trial can have only two possible outcomes—success or failure. 2. There must be a fixed number of trials. 3. The outcomes of each trial must be independent of each other. 4. The probability of success must remain the same for each trial. © Copyright Mc. Graw-Hill 2004 14
The Binomial Experiment (cont’d. ) n The outcomes of a binomial experiment and the corresponding probabilities of these outcomes are called a binomial distribution. © Copyright Mc. Graw-Hill 2004 15
Notation for the Binomial Distribution n The symbol for the probability of success n The symbol for the probability of failure n The numerical probability of success n The numerical probability of failure and The number of trials n The number of successes © Copyright Mc. Graw-Hill 2004 16
Binomial Probability Formula n In a binomial experiment, the probability of exactly X successes in n trials is © Copyright Mc. Graw-Hill 2004 17
Binomial Distribution Properties The mean, variance, and standard deviation of a variable that has the binomial distribution can be found by using the following formulas. mean variance standard deviation © Copyright Mc. Graw-Hill 2004 18
Other Types of Distributions n The multinomial distribution is similar to the binomial distribution but has the advantage of allowing one to compute probabilities when there are more than two outcomes. n The multinomial distribution is a general distribution, and the binomial distribution is a special case of the multinomial distribution. © Copyright Mc. Graw-Hill 2004 19
Homework n Homework Chapter 5, Review Exercises Page 276 Ø 1, 2, 3, 8, 10, 12, 16, 17 © Copyright Mc. Graw-Hill 2004 20
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