(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the MGF andallthe moments for [itex]X\sim N(0,1)[/itex]

2. The attempt at a solution

For the MGF, I have:

[tex]M_X(s)=\displaystyle\int_{-\infty}^{\infty}e^{sx}\frac{e^{x^2/2}}{\sqrt{2\pi}}\,dx = \ldots=e^{s^2/2}[/tex]

Next I found that:

[tex]M'_X(0)=E[X]=0[/tex]

[tex]M''_X(0)=E[X^2]=1[/tex]

[tex]E[X^3]=0[/tex]

[tex]E[X^4]=3[/tex]

[tex]\ldots[/tex]

[tex]E[X^{ODD}]=\{0\}[/tex]

[tex]E[X^{EVEN}]=\{1,3,15,105,945,\ldots\}[/tex]

Is it enough to write:

[tex]E[X^k]=M_X^{(k)}(0)=\frac{d^k}{ds^k}e^{s^2/2}[/tex]

Am I totally off track here? How would I prove this?

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# Homework Help: MGF and moments

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