LEC19 Mean & Var of n rv

# LEC19 Mean & Var of n rv - Mean and Variance of a...

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Mean and Variance of a Linear Function of n independent random variables Given: X i , i = 1. ..n, independent random variables E(X i ), V(X i ) i=1. ..n Linear function Y c c X c X n n = + + + 0 1 1 .... The mean and variance of Y: E Y c c E X c E X V Y c V X c V X n n n n ( ) ( ) .... ( ) ( ) ( ) .... ( ) = + + + = + + 0 1 1 1 2 1 2 X i 's are independent! If X i 's are Normal, then Y is Normal too.

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Example XYZ sells 2 products. The amounts sold are independent. The amount of product 1 sold each month has mean 1.4 and variance .24. The profit per unit is \$6. The amount of product 2 sold each month has mean 1.7 and variance .21. The profit per unit is \$4. Find the mean and standard deviation of the monthly profit. 2
r.v. X i = number units of product i sold per month, i=1,2 r.v. P = profit per month r.v. P = 6X 1 + 4X 2 since P is a linear function of X 1 and X 2 and since X 1 and X 2 are independent: E P E X E X month V P V X V X SD P month ( ) ( ) ( ) ( . ) ( . ) \$15. /

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LEC19 Mean & Var of n rv - Mean and Variance of a...

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