Chapter 5 Discrete Distributions

5.7. FUNCTIONS OF DISCRETE RANDOM VARIABLES 135
1. f (x) = Cx n , 0 < x < 1.
2. f (x) = Cxe −x , 0 < x < ∞.
3. f (x) = e −(x−C) , 7 < x < ∞.
4. f (x) = Cx 3 (1 − x) 2 , 0 < x < 1.
5. f (x) = C(1 + x 2 /4) −1 , −∞ < x < ∞.
Exercise 5.4. Show that IE(X − µ) 2 = IE X 2 − µ 2 . Hint: expand the quantity (X − µ) 2 and
distribute the expectation over the resulting terms.
Exercise 5.5. If X ∼ binom(size = n, prob = p)showthatIEX(X − 1) = n(n − 1)p 2 .
Exercise 5.6. Calculate the mean and variance of the hypergeometric distribution. Show that
µ = K M
M + N , σ2 = K MN
(M + N) 2 M + N − K
M + N − 1 . (5.7.4)