Stat 331/Elec 331, Homework 3, October 2
Solutions should be clear and easy to follow. You are allowed to use the
book and lecture notes. Boldface numbers within parentheses denote the
maximum score on each problem.
Solutions are due on the date at the top. If you can not come to class and
hand it to me there, you will have to come by my office (slide it under the
door if I am not there). If you can not make it on time, you may still return
your solutions but there will be a two point deduction for each day you are
late.
1. Let R∼unif [0,1] and let Vbe the volume of a sphere with radius R.
Compute E[V] and Var[V]. (3)
2. If Xis a continuous random varible, the median of Xis defined as the
value mwhich is such that P(X≤m) = P(X≥m) = 1/2, in some sense
the ”midpoint” of the distribution. In the following cases, find the median
mand compare with the mean µ.
a. X∼unif[a, b].
b. X∼exp(a).
c. Xhas pdf f(x) = 1/x2, x ≥1. (3)
3. The non-negative continuous random variable Xhas failure rate function
h(t) = 1/(1 + t), t ≥0. Find the pdf of X(3)
4. Let X∼unif(0,1). Find the failure rate function h(t) of X. What hap-
pens to h(t) as t→1? Explain intuitively. (3)
5a. Let Xbe a discrete random variable with range {1,2, ...}. The (discrete)
