- #1
spitz
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Homework Statement
Find the MGF and all the 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?