Calculating the mean and variance from a moment generating function

[Charlotte87]

Homework Statement

Assume that X is squared-Chi-distributed, which means that the moment generating function is given by:

$m(t)=(1-2t)^{-k/2}$

Use the mgf to find E(X) and var(X)

The Attempt at a Solution

I know that m'(0)=E(X), and m''(0)=var(X).

So I find:

$m'(t)=k(1-2t)^{-(k/2)-1}$
which gives m'(0)=k

Similarily, I find

$m''(t)=(k^{2}+2k)(1-2t)^{-(k/2)-2}$
which gives m''(0)=k^2+2k

However, in my textbook, it says that the variance of a square-chi distribution should be 2k, not k^2. Where do I go wrong?

 

[LCKurtz]

Homework Statement

Assume that X is squared-Chi-distributed, which means that the moment generating function is given by:

$m(t)=(1-2t)^{-k/2}$

Use the mgf to find E(X) and var(X)

The Attempt at a Solution

I know that m'(0)=E(X), and m''(0)=var(X).

Your mistake is right there. m''(0) = E(X²), not var(X)

So I find:

$m'(t)=k(1-2t)^{-(k/2)-1}$
which gives m'(0)=k

Similarily, I find

$m''(t)=(k^{2}+2k)(1-2t)^{-(k/2)-2}$
which gives m''(0)=k^2+2k

However, in my textbook, it says that the variance of a square-chi distribution should be 2k, not k^2. Where do I go wrong?

 

[Charlotte87]

Of course. Then var(X)=E(X^2)-(E(X))^2 =k^2+2k-k^2=2k.

Thank you!