- #1
Mark53
- 93
- 0
Homework Statement
[/B]
Let X be a random variable with support on the positive integers (1, 2, 3, . . .) and PMF f(x) = C2 ^(-x) .
(a) For what value(s) of C is f a valid PMF?
(b) Show that the moment generating function of X is m(t) = Ce^t/(2− e^t) , and determine the interval for t for which it is valid. (You may use your value for C calculated in question 1, if you would like).
(c) Using the MGF, calculate the expected value and the variance of X.
The Attempt at a Solution
[/B]
a)
sum from -∞ to ∞ of C/(2^x)=1
C(1/2+1/4...)=1
C=1 as it converges
b)
m(t)=E[e^tx]=integral from -∞ to ∞ of ((e^tx)*(2^(-x)))
=integral from -∞ to ∞ of (e^tx)/(2^x)
is this the right way to go about calculating it?