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Let X~N(0,1)m find the moment generating function of $X^2$ using integration techniques.

I'm not sure exactly what this is asking me to do. Is $X^2$ just the pdf for the standard normal function squared? Any pointers appreciated!

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Is $X^2$ just the pdf for the standard normal function squared?

If you mean the pdf of $X^2$, then no. $X^2$ will follow a $\chi^2$ distribution with one degree of freedom. You can't find pdfs of transformed variables simply by applying the transformation to the pdf of the original variable.

That's not important, though. The definition of a moment generating function is given by an expectation:

$$\mathbb E[e^{tX^2}].$$ You can find this by integration just like you would any other expectation of a function of $X$:

$$\mathbb E[e^{tX^2}] = \int e^{tx^2} f_X(x)\, dx,$$ where $f_X(x)$ is the standard normal pdf.

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