1. (10 pts) Let X be a continuous random variable with the following probability density function
(PDF),
f(x) =
1 if 0 ≤ x < 1
2
1
3
x
−3
if 1
2 ≤ x ≤ 1
0 otherwise
1. Find the cumulative distribution function (CDF) of X.
2. Find the 90th percentile of the distribution.
1
2. (10 pts) Let X1, X2, . . . , X100 be a random sample from a Pareto distribution with PDF
f(x) =
3x
−4
if x ≥ 1
0 otherwise.
Find normal approximations for each of the following.
1. P
P100
i=1 Xi > 160
.
2. The 90th percentile of the sample mean Xn =
1
100
P100
i=1 Xi
.
3. (10 pts) Consider independent and identically distributed random samples, X1, . . . , Xn of size
n from a distribution with PDF f(x) = (α − 1)/xα
if 1 ≤ x < ∞; zero otherwise, where α > 1.
Let Yn = min{X1, . . . , Xn}.
1. Find the PDF of the smallest order statistic Yn.
2. Find the CDF of the smallest order statistic Yn.
3. What is the limiting distribution for Yn, as n → ∞?
4. Find the limiting distribution for Y
n
n
, as n → ∞ (if it exists).
3
4. (10 pts) Let (X, Y ) be a vector of two continuous random variables, with the following joint
probability density function (joint PDF),
f(x, y) =
e
−3y
, if 0 ≤ x, y < ∞ and y ≥ x;
0, otherwise.
1. Find the joint PDF of the variables S = X + Y and T = X.
2. Find the marginal PDF of S.