(a) Suppose that X1 ~ N(μ1, Ï12) and X2 ~ N(μ2, Ï22) are independently distributed. What is...

(a) Suppose that X1 ~ N(μ1, σ12) and X2 ~ N(μ2, σ22) are independently distributed. What is the variance of
Y = pX1 + (1 - p)X2?
Show that the variance is minimized when

What is the variance of Y in this case?
(b) More generally suppose that Xi ~ N(μi, σi2), 1 ‰¤ i ‰¤ n, are independently distributed, and that
Y = p1X1 + ... + pnXn
where p1 + ... + pn = 1. What values of the pi minimize the variance of Y, and what is the minimum variance?

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