Homework 7

# Homework 7 - University of California Los Angeles...

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University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Homework 7 EXERCISE 1 Show that V ar ( X - Y ) = V ar ( X ) + V ar ( Y ) - 2 Cov ( X,Y ). EXERCISE 2 If X and Y are independent variables with equal variances ﬁnd Cov ( X + Y,X - Y ). EXERCISE 3 If U = a + bX and V = c + dY . show that | ρ UV | = | ρ XY | . EXERCISE 4 Let U and V be independent random variables with means μ and variances σ 2 . Let Z = αU + V 1 - α 2 . Find E ( Z ) and ρ UZ . EXERCISE 5 Suppose that X and Y are two independent measurements. Also it is given that E ( X ) = E ( Y ) = μ , but σ X and σ Y are unequal. The two measurements are combined by means of a weighted average to give Z = αX + (1 - α ) Y where α is a scalar and 0 α 1. a. Show that E ( Z ) = μ . b. Find α in terms of σ X and σ Y to minimize V ar ( Z ). c. Under what circumstances is it better to use the average X + Y 2 than either X or Y alone to estimate μ . Note: X,Y, X

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Homework 7 - University of California Los Angeles...

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