SID:
1. (10 pts) Part (a) and (b) of this question are independent.
(a) (5 pts) Suppose
a random variable U follows Uniform(0,1) distribution. Define
U
W = log 1−U
. Here log is log base e.
i. (2 pts) Find the range of W .
ii. (3 pts) Find the density of W .
(b) (5 pts) The joint distribution of two random variable X and Y is
fX,Y (x, y) = 8 x y 1(0<y<x<1) .
Here 8 x y 1(0<y<x<1) means 8 x y when 0 < y < x < 1 and zero otherwise.
Y
Define Z = X
.
i. (1 pt) Find the range of X. Find the range of Z.
ii. (1 pt) Find the joint density gX,Z (x, z) of X and Z. Hint: Use the change
of variable formula for (X, Y ) to (X, Z(X, Y )) in Lec. 31.
iii. (1 pt) Find the marginal density of X.
iv. (1 pt) Find the marginal density of Z.
v. (1 pt) Are X and Z independent?
2
SID:
2. (10 pts) In this problem you will use the MGF to prove that the square of the
standard normal is a well known distribution.
2
(a) (3 pts) Let Z ∼ N (0, 1) (recall fZ (z) = √12π ez ). Using the definition of MGF
show that
Z ∞
1
1
2
√ e− 2 (1−2t)z dz
MZ 2 (t) =
2π
−∞
(b) (5 pts) Using your result from part (a) show that
1
2
MZ 2 (t) =
1
2
−t
21
, for t <
Hint: The density of Z ∼ N (0, σ 2 ) is f (z) =
1
√ 1 e− 2
2πσ
1
2
z2
σ2
.
(c) (2 pts) Make a conclusion about the distribution of Z 2 . For full credit explain
your reasoning.
3
SID:
3. (10 pts) Suppose you throw 10 darts at a Cartesian plane, with X and Y coordinates
of the dart following a standard normal distribution, N (0, 1). Each dart throw is
independent of each other.
(a) (5 pts) Find the probability that each dart is between 3 units and 5 units away
from the origin of the Cartesian plane.
(b) (5 pts) Let D1 , ..., D10 be the distance each dart is away from the origin of the
Cartesian plane. Find the density of D(k) where D(k) is the k th order statistic.
4
SID:
4. (10 pts) Let X, Y, Z ∼ N (0, 1) be independent.
(a) (5pts) Find P (X − 2Y + 3Z > 4).
(b) (2pts) Draw the region
1
min(X, Y ) < √ max(X, Y )
3
in the plane using the space provided.
Hint: In the region X < Y in the plane find where X <
Y < X in the plane find where Y < √13 X.
(c) (3pts) Using part (b) above, find P (min(X, Y ) <
5
√1
3
√1 Y
3
and in the region
max(X, Y )).
SID:
5. (10 pts) Let X, Y be iid random variables each with density fX (x) =
and zero otherwise.
(a) (5 pts) Find the density of Z = Y /X.
√
(b) (5 pts) Find E[ Z].
6
1
x2
for x > 1