Moment generating functions help

[Mark53]

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?

 

[Ray Vickson]

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?

No. For a discrete random variable we have
$$E f(X) = \sum_x p(x) f(x), $$
so involves summation, not integration.

 

[Mark53]

No. For a discrete random variable we have
$$E f(X) = \sum_x p(x) f(x), $$
so involves summation, not integration.

when calculating the sum I get:

the sum of x=0 to ∞ of (e^tx)/(2^x)=e^t/2

which is wrong am I still missing something?

 

[Ray Vickson]

when calculating the sum I get:

the sum of x=0 to ∞ of (e^tx)/(2^x)=e^t/2

which is wrong am I still missing something?

Yes: you are basically saying that $\sum_{k=0}^{\infty} r^k = r,$ which is wrong.

 

[Mark53]

Yes: you are basically saying that $\sum_{k=0}^{\infty} r^k = r,$ which is wrong.

Do I need to see if the series converges or find the partial sum?

not sure how to start solving it