Download EXERCISES II 1. Let X, Y be independent identically distributed

EXERCISES II
1. Let X, Y be independent identically distributed random variables with density function f .
(a) Compute P(X ≥ Y ).
(b) Let
X(1) := min{X1 , X2 },
X(2) := max{X1 , X2 }.
Compute the distribution functions F (i) (x) := P(X(i) ≤ x), for i = 1, 2.
(c) Compute the density functions of X(i) , i = 1, 2 and check that they are indeed valid density
functions.
2. Let X be a Poisson random variable with parameter λ
P(X = k) =
eλ λk
,
k!
k = 0, 1, 2, . . . .
(a) Find its MGF.
(b) Use the MGF to
(i) find the mean and variance of X;
(ii) Show that if X1 and X2 are independent Poisson random variables with parameters λ1
and λ2 respectively then Z = X1 + X2 is also Poisson with parameter λ1 + λ2 .
(c) Do question b.(ii) but without using the MGF.
3. Let (X, Y ) be jointly normal with probability density function
fX,Y (x, y) =
1
2π 1 − ρ2 σX σY
n (x − µ )2
h
1
(y − µY )2 2ρ(x − µX )(y − µY ) oi
X
exp −
+
−
.
2
2(1 − ρ2 )
σX σY
σX
σY2
p
(a) Compute the marginals fX , fY . What is the distribution of X? Of Y ?
(b) Compute the conditional density fX|Y (x|y). What is the distribution of Y given X = x?
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