- #1

senobim

- 34

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I have calculated characteristic function of normal distribution [tex]f_{X}(k)=e^{(ika-\frac{\sigma ^{2}k^{2}}{2})}[/tex] and now I would like to find the moments, so I know that you could expand characteristic function by Taylor series

[tex]f_{X}(k)=exp(1+\frac{1}{1!}(ika - \frac{\sigma^2k^2}{2})+\frac{1}{2!}(ika - \frac{\sigma^2k^2}{2})^2+\frac{1}{3!}(ika - \frac{\sigma^2k^2}{2})^3+...)[/tex]

[tex]f_{X}(k)=exp(1+\frac{(ik)}{1!}\left \langle X^1 \right \rangle+\frac{(ik)^2}{2!}\left \langle X^2 \right \rangle+\frac{(ik)^3}{3!}\left \langle X^3 \right \rangle+...)[/tex]

and the moments will be

[tex]\left \langle X^n \right \rangle[/tex]

Now the problem is that I completely forgot how to evaluate Taylor series.

Could you help me to calculate for example second moment? I know what the answer should be, but I couldn't get it right.

[tex]f_{X}(k)=exp(1+\frac{1}{1!}(ika - \frac{\sigma^2k^2}{2})+\frac{1}{2!}(ika - \frac{\sigma^2k^2}{2})^2+\frac{1}{3!}(ika - \frac{\sigma^2k^2}{2})^3+...)[/tex]

[tex]f_{X}(k)=exp(1+\frac{(ik)}{1!}\left \langle X^1 \right \rangle+\frac{(ik)^2}{2!}\left \langle X^2 \right \rangle+\frac{(ik)^3}{3!}\left \langle X^3 \right \rangle+...)[/tex]

and the moments will be

[tex]\left \langle X^n \right \rangle[/tex]

Now the problem is that I completely forgot how to evaluate Taylor series.

Could you help me to calculate for example second moment? I know what the answer should be, but I couldn't get it right.

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