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# Lecture7_print - MA/STAT416 Probability Lecture Notes...

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MA/STAT416: Probability Lecture Notes Spring 2011 1 Chapter 4: Discrete Random Variables and Mass Functions February 28, 2011 4 Expectation 4.4 Mean and Variance Definition 1. Let X be a discrete random variable taking the values x 1 , x 2 , · · · , x n . Furthermore, let p 1 , p 2 , · · · , p n be the pmf of X . The expected value of X is defined by E ( X ) = n i =1 x i · p i . Note: The mean of X is μ X = E ( X ). Definition 2. The variance of X is defined by σ 2 X = V ar ( X ) = n i =1 ( x i - μ X ) 2 · p i . Note: The variance of X is V ar ( X ) = E ( X 2 ) - [ E ( X )] 2 = E ( X 2 ) - μ 2 X . Example 3 . For each of the following random variables, find the mean and variance. 1. X = number of heads in 2 tosses of a fair coin. 2. X = sum of two fair dice rolls. 3. X = larger of two numbers chosen with replacement from 1 , 2 , · · · , 10. Property: Let X 1 , X 2 be two random variables such that X = X 1 + X 2 . Then, E ( X ) = E ( X 1 + X 2 ) = E ( X 1 ) + E ( X 2 ). Example 4 . 3 cookies are distributed completely at random (and independently) to 3 kids. Let X be the number of kids who do not receive any cookies. Find the expected

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Lecture7_print - MA/STAT416 Probability Lecture Notes...

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