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Unformatted text preview: 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 X = E ( X ). Definition 2. The variance of X is defined by 2 X = V ar ( X ) = n i =1 ( x i- X ) 2 p i . Note: The variance of X is V ar ( X ) = E ( X 2 )- [ E ( X )] 2 = E ( X 2 )- 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 )....
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