Let f1(y) and f2(y) be density functions and let a be a constant such that 0 a 1

Chapter 4, Problem 4.185

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Let f1(y) and f2(y) be density functions and let a be a constant such that 0 a 1. Consider the function f (y) = a f1(y) + (1 a) f2(y).a Show that f (y) is a density function. Such a density function is often referred to as a mixture of two density functions. b Suppose that Y1 is a random variable with density function f1(y) and that E(Y1) = 1 and Var(Y1) = 2 1 ; and similarly suppose that Y2 is a random variable with density function f2(y) and that E(Y2) = 2 and Var(Y2) = 2 2 . Assume that Y is a random variable whose density is a mixture of the densities corresponding to Y1 and Y2. Show that i E(Y ) = a1 + (1 a)2. ii Var(Y ) = a2 1 + (1 a)2 2 + a(1 a)[1 2] 2. [Hint: E(Y 2 i ) = 2 i + 2 i , i = 1, 2.]