A few more practice questions for the ﬁnal exam
For the ﬁnal exam you are responsible for all the material we have covered in the course. This document
contains some practice questions on the later material (after the midterm).
1. Suppose X∼Exp(1). Derive the distribution of Y=eX2using:
(a) The transformation (change of variable) technique.
(b) The distribution function technique.
2. Suppose X∼Unif(−1,3). What is the distribution of Y=|X|?
3. Suppose the random variable Xhas the probability density function:
f(x;α, β) =
βαxα−1e−(x/β)αfor x > 0
where α > 0, β > 0. What is the distribution of Y= (X/β)α?
4. Show that if Xhas a geometric distribution, P(X=b+c|X > b) = P(X=c).
5. In class and the notes we deﬁned a geometric random variable (X, say) to be the number of
trials required to get the ﬁrst success in repeated independent Bernoulli trials. We learned that
E(X) = 1
p,V ar(X) = 1−p
p2, and MX(t) = pet
1−(1−p)et(for t < −ln(1 −p)).
Many sources deﬁne a geometric random variable (Y, say) to be the number of failures before
getting the ﬁrst success. Suppose we want to work with this alternative deﬁnition.
(a) What is the relationship between Xand Y?
(b) What is the probability mass function for the random variable Y?
(c) What is the support of this probability mass function?
(d) What is the mean of Y?
(e) What is the variance of Y?
(f) What is the moment generating function of Y?
6. Hanhart syndrome is a rare genetic disorder that aﬀects approximately one in 15,000 live births in
North America. Individuals with Hanhart syndrome can have a number of physical deformities,
including malformed ﬁngers, toes, and limbs. Suppose 10,000 North American live births are
(a) Using the binomial distribution, what is the probability that at least one individual has
(b) Using the Poisson approximation to the binomial distribution, what is the probability that
at least one individual has Hanhart syndrome?
7. Give the probability mass function for the random variable in the following situations, including
the value(s) of the parameter(s) and the support of the function.
(a) The number of hearts if 10 cards are dealt without replacement from a standard deck.
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