ST5201 AY 2008/2009
Examples and Exercise 4
1. Consider X ∼ P oisson(λ), i.e.
P (X = x) = e−λ
λx
x!
for x = 0, 1, 2, . . . .
(a) Find the Moment Generating Function of X.
(b) Find E(X) and V ar(X).
2. The pair of random variables (X, Y ) has the distribution.
Y
2
3
4
1
X
2
3
1/12
1/6
0
1/6
0
1/3
1/12
1/6
0
(a) Find E(X), var(X).
(b) Find cov(X, Y ).
3. Suppose the distribution of Y conditional on X = x is normal N (x, x2 ) and the marginal distribution
of X is uniform on [0, 1].
(a) Find E(Y ), var(Y ) and cov(X, Y ).
(b) Prove that Y /X and X are independent.
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ST5201, AY 2008/09
Examples and Exercise 4
4. (a) Let Z be a standard normal random variable N (0, 1), show that the MGF of Z is
MX (t) = exp(t2 /2).
(b) Let X and Y be two independent standard normal random variables. Find the MGF of
U =X +Y.
(c) Let X be a Gamma distribution with pdf
fX (x) =
Show that the MGF of X is
λα α−1 −λx
x
e
, x ≥ 0.
Γ(α)
t −α
.
MX (t) = 1 −
λ
(d) Consider Z be a standard normal random variable N (0, 1), find the MGF of Z 2 .
(e) Let X and Y be two independent standard normal random variables. Find the MGF of
U = X 2 + Y 2.
(f) Extend the result to MGF of U = X12 +X22 +· · ·+Xn2 , where X1 , X2 , . . . , Xn are n independent
N (0, 1) random variables.
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2008