Suppose that X is a r.v. whose moment generating function is: m(theta) = 1/5e^theta 2/5e^(4theta) 2/5e^(8theta), theta from negative infinity to infinity. Find the values (range) of X and its probability distribution Solution From the moment generating function, it can be easily recognized that X is a discrete distribution with 3 possible value 1, 4 and 8 associated with the possibilities of 1/5, 2/5 and 2/5 respectively; X=1 with p=1/5, X=4 with p=2/5, X=8 with p=2/5..

1. Suppose that X is a r.v. whose moment generating function is: m(theta) = 1/5e^theta
2/5e^(4theta) 2/5e^(8theta), theta from negative infinity to infinity. Find the values (range) of X
and its probability distribution
Solution
From the moment generating function, it can be easily recognized that X is a
discrete distribution with 3 possible value 1, 4 and 8 associated with the possibilities of 1/5, 2/5
and 2/5 respectively; X=1 with p=1/5, X=4 with p=2/5, X=8 with p=2/5.