Math 419 – Spring 2016
MGF
Problem Set 10
Moment Generating Functions
1. Find the moment generating function for the random variable with probability density function
f(x) = (1/(e2 – 1) ) e1 – x , for 1 £ x £ 3.
2. Given f(x) = xe – x, x > 0, what is the moment generating function?
Math 419 – Spring 2016
MGF
Problem Set 10
3. Let X and Y be independent and identically distributed random variables with a common moment
generating function M(t) for which M '(0) = 3 and M''(0) = 25. Calculate the variance of X + 2Y.
4. Let X be an exponential random variable with mean 2. Calculate E[100(0.5)X].
Math 419 – Spring 2016
MGF
Problem Set 10
5. Find the MGF for the random variable with probability mass function P(N = 1) = ½, P(N = 2) = ¼ ,
P(N = 3) = ¼ , and zero otherwise.
6. Let X be the value of the bill drawn at random from a bag containing FIVE $1 bills, TWO $5 bills,
and THREE $10 bills. Calculate the skewness of X.
Math 419 – Spring 2016
MGF
Problem Set 10
7. Given the moment generating function M(t) = ((1 + 2e3t )/3 )4, find the standard deviation.
Math 419 – Spring 2016
Properties of Joint MGFs
MGF
Problem Set 10
Math 419 – Spring 2016
MGF
Problem Set 10
8. The joint moment function for X and Y is MX, Y (t1, t2) = 1/(1 – t1 – t2 – t1t2) , where t1 < 1, t2 < 1. Find
the pdf of X.
Math 419 – Spring 2016
MGF
9. Let MX, Y (s, t) = e 4s^2 + st + 8t^2. Find the coefficient of correlation of X and Y.
Problem Set 10
Math 419 – Spring 2016
MGF
Problem Set 10
10. A company insures homes in three cities, J, K, and L. Since sufficient distance separates the cities,
it is reasonable to assume that the losses occurring in these cities are independent. The moment
generating functions for the loss distributions of the cities are MJ(t) = (1 – 2t) – 3,
MK(t) = (1 – 2t) – 2.5, ML(t) = (1 – 2t) – 4.5, Let X represent the combined losses from the three cities.
Calculate E[X 3].