How do I find the moment generating function of $N(0, 1)$

Question

Here is my homework, I just don't really know how to find the moment generating function. I can do the rest after i get that. Any help?

Answer 1

hint A moment generating function is an expectation value you must take in one way or another. Here you want $$E(e^{tZ^2}) = \int_{-\infty}^\infty e^{tz^2}\frac{e^{-z^2/2}}{\sqrt{2\pi}}dz= \int_{-\infty}^\infty \frac{e^{-\frac{1}{2}(1-2t)z^2}}{\sqrt{2\pi}}dz.$$ This integral is simple to do via a substitution, and to generalize to $Z_1^2+\ldots +Z_n^2.$ (The MGF of the sum of $n$ iids is just the MGF raised to the n-th power.)

Answer 2

Moment generating function of $X\sim N(0,1)$ is by definition $$M_X(t)=\int_{-\infty}^{+\infty} e^{tx} \frac 1 {\sqrt{2\pi}} e^{-\frac{x^2}2} \, dx=\int_{-\infty}^{+\infty} \frac 1 {\sqrt{2\pi}} e^{-\frac{x^2-2tx}2} \, dx=\int_{-\infty}^{+\infty} \frac 1 {\sqrt{2\pi}} e^{-\frac{(x-t)^2+t^2}2} \, dx=$$ $$=e^{\frac{t^2}2}\int_{-\infty}^{+\infty} \frac 1 {\sqrt{2\pi}} e^{-\frac{(x-t)^2}2} \, dx=e^{\frac{t^2}2}$$ since the final integral equals one (why?).

But you actually need to find the moment generating function of $X^2$, so you need $$M_{X^2}(t)=\int_{-\infty}^{+\infty} e^{tx^2} \frac 1 {\sqrt{2\pi}} e^{-\frac{x^2}2} \, dx.$$ Try to put both exponentials in one and try a change of variables.

You can also calculate $F_{X^2}(t)=P(X^2\le t)$. First consider the cases $t<0$ and $t=0$, which will return zero probability, and then work the case $t>0$ to relate $F_{X^2}$ and $F_X$. You can then take derivatives to put the relation in terms of densities instead of cumulative distributions.

That is: try to prove that for $t>0$ $$F_{X^2}(t)=2 F_X(\sqrt t)-1.$$

If you take a derivative (with respect to $t$; I then evaluated in $t=x$ just for convention), it turns out that a pdf for $X^2$ is given by $$f_{X^2}(x)=2 f_X(\sqrt x)\cdot \frac 1 {2\sqrt x},\quad x>0,$$ (and $0$ otherwise).

Now you just have to substitute $f_X$ by its actual formula to find that this is the density of a $\Gamma$ distribution for some specific parameter values.