Answer to 5.3.3
thanhnga la (tla@ucla.edu)
Sun, 05 Nov 95 22:33:29 -0800
To calculate Var(w2), first we need to calculate the cdf Fw2(w) as
FW2(w) = P(w2 <= w) = P(n*Ymin <= w) = P(Ymin <= w/n)
= 1 - P(Ymin > w/n)
(Ymin > w/n) ~ (Y1 > w/n, Y2 > w/n,...,Yn > w/n) because all Yi > Ymin
Then, FW2(w) = 1 - P(Y1 > w/n, Y2 > w/n,...,Yn > w/n)
= 1 - (P(Y1 > w/n) * P(Y2 > w/n)*...*P(Yn > w/n))
= 1 - (P(Y1 > w/n))^n
= 1 - (integral(from w/n to infinity) of fY(y; theta))^n
= 1 - e^(-w/theta)
take derivative of the cdf to get the pdf
fW2(w;theta) = derivative of FW2(w) = (1/theta)*e^(-w/theta)
Find Var(W2) based on this fW2 but notice that this exponential pdf
looks like the given fY(y;theta), only it is w instead of y. If you can
get Var(Y1) in the first question which is (theta)^2, then you can get
the same value for Var(W2).
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