# Ch07 - Chapter 7 Random Variables and Discrete probability...

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1 Random Variables and Discrete probability Distributions Chapter 7

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2 7.2 Random Variables and Probability Distributions A random variable is a function or rule that assigns a numerical value to each simple event in a sample space. A random variable reflects the aspect of a random experiment that is of interest for us. There are two types of random variables: Discrete random variable Continuous random variable.
3 A random variable is discrete if it can assume a countable number of values. A random variable is continuous if it can assume an uncountable number of values. 0 1 1/2 1/4 1/16 Continuous random variable After the first value is defined the second value, and any value thereafter are known. Therefore, the number of values is countable After the first value is defined, any number can be the next one Discrete random variable Therefore, the number of values is uncountable 0 1 2 3 . .. Discrete and Continuous Random Variables

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4 A table, formula, or graph that lists all possible values a discrete random variable can assume, together with associated probabilities, is called a discrete probability distribution . To calculate the probability that the random variable X assumes the value x, P( X = x), add the probabilities of all the simple events for which X is equal to x, or Use probability calculation tools (tree diagram), Apply probability definitions Discrete Probability Distribution
5 If a random variable can assume values x i , then the following must be true: 1 ) x ( p . 2 x all for 1 ) p(x 0 . 1 i x all i i i = Requirements for a Discrete Distribution

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6 Distribution and Relative Frequencies In practice, often probability distributions are estimated from relative frequencies. Example 7.1 A survey reveals the following frequencies (1,000s) for the number of color TVs per household. Number of TVs Number of Households x p(x) 0 1,218 0 1218/Total = .012 1 32,379 1 .319 2 37,961 2 .374 3 19,387 3 .191 4 7,714 4 .076 5 2,842 5 .028 Total 101,501 1.000
7 Determining Probability of Events The probability distribution can be used to calculate the probability of different events Example 7.1 – continued Calculate the probability of the following events: P(The number of color TVs is 3) = P(X=3) =.191 P(The number of color TVs is two or more)

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8 Probability calculation techniques can be used to develop probability distributions Example 7.2 A mutual fund sales person knows that there is 20% chance of closing a sale on each call she makes. What is the probability distribution of the number of sales if she plans to call three customers? Developing a Probability Distribution
9 Solution Use probability rules and trees Define event S = {A sale is made}. Developing a Probability Distribution P(S)=.2 P(S C )=.8 P(S)=.2 P(S)=.2 P(S)=.2 P(S)=.2 P(S C )=.8 P(S C )=.8 P(S C )=.8 P(S C )=.8 S S S S S S C S S S S S S C S S S S S S C S S S S S S C P(S)=.2 P(S C )=.8 P(S C )=.8 P(S)=.2 P(x) .2 3 = .008 3(.032)=.096 3(.128)=.384 .8 3 = .512 (.2)(.2)(.8)= .032 Probability Distribution Finding Probability/ Probability Distribution

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Ch07 - Chapter 7 Random Variables and Discrete probability...

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