Let X1 and X2 be two continuous random variables with density functions f1 and f2, expected values ????1 and ????2 and variances ????2 1 and ????2 2 , respectively. Consider now a function f defined by f (x) = ????f1(x) + (1 − ????)f2(x), x ∈ ℝ, where 0 ≤ ???? ≤ 1.

(i) Verify that f is the density function of some random variable X.

(ii) Show that the expected value and the second moment around zero of X are given by the formulas

???? = E(X) = ????????1 + (1 − ????)????2, E(X2) = ????E(X2 1) + (1 − ????)E(X2 2 ).

(iii) Deduce from above an expression for the variance of X in terms of ????, ????1, ????2, ????2 1, ????2 2 .