STAT 400
Exam 2 Practice Problems
1. A continuous random variable Yhas a cumulative distribution function given by
F(y) =
0y < 0
y20≤y < 1
1y≥1
(a) Find P(1
2< Y ≤3
4) by using the cumulative distribution function.
(b) Find P(1
2< Y ≤3
4) by using the probability density function.
2. Let Xbe a continuous random variable distributed χ2(12)
(a) Fincd E(X) and V ar(X).
(b) Let X1, X2, X3be independent χ2(12) random variables. What is the distribution of
Y=X1+X2+X3?.
3. Suppose that math SAT scores for a particular population are normally distributed
with mean 475 and a standard deviation of 100.
(a) Find P[Score >650].
(a) What fraction of the population scores between 600 and 750?
(b) What score marks the 75th percentile?
(c) Given a random sample of size n= 10 scores, what is the distribution of the sample
mean?
4. Let X1and X2be two independent raondom variables with respective means 3 and 7
and variances 9 and 25. Compute the mean and variance of Y=−2X1+X2.
5. Let Xequal the outcome when a four-sided die is rolled. Let Yequal the outcome
when a six-sided die is rolled. Assume Xand Yare independent. Let W=X+Y.
(a) Find the moment-generating function of W.
(b) Give the probability mass function of W.
6 Let Xbe the mean of a random sample of size n= 36 from an exponential distribution
with mean 3. Use the Central Limit Theorem to approximate P(2.5≤X≤4).