Download Moment Generating Function Homework Problems – Extra Credit 1

Moment Generating Function Homework Problems – Extra Credit
1. The probability distribution for a random variable X is given by
P(X=0) = 1/2, P(X=1) = 1/4, P(X=2) = 1/8, and P(X=3) = 1/8.
a) Find the moment generating function for X.
b) Determine the first two moments for X from the moment generating function.
c) Find the variance of X from the moments.
2. Consider the Bernoulli Random Variable with P(X=1)=p and P(X=0) =1-p = q.
a) Show that the Moment generating function for X is
.
b) Determine the first two moments for X from the moment generating function.
c) Find the variance of X from the moments.
d) Consider a binomial random variable Y as the sum of n Bernoulli trials. That is, consider
. Using the rule for moment generating functions of sums, provide
another way to find the moment generating function for the binomial distribution Y.
3. Suppose that X is a Geometric Distribution. That is suppose that P(X=x) = pq
a) Show that the moment generating function for X is
x-1
, for x=1,2, …
.
b) Determine the first two moments for X from the moment generating function.
c) Find the variance of X from the moments.
4. Suppose that X has an exponential distribution with parameter λ. That is suppose that
.
a) Prove that the moment generating function for X is (1 − t/ λ)-1 .
b) Determine the first two moments for X from the moment generating function.
c) Find the variance of X from the moments.
5. Suppose that the number of players experiencing concussions is estimated to be 3 per year,
the number of players needing stitches is estimated to be 4 per year, and the number of players
breaking a leg is estimated to be 2 per year. Assume that each of these random variables X1, X2,
and X3 representing the number of players with concussions, the number of players who
needed stitches, and the number of players who broke a leg, respectively, is independent.
Assume also that each of these random variables follows a Poisson distribution. Find the
probability that fewer than 3 of these types of accident occur in one year. (Hint: Think about
the random variable
. Remember what we proved about the Reproductive
Property of Poisson Distribution. )