Density function from moment generating function

Asked Apr 26 '20 at 07:36

Active Apr 26 '20 at 07:36

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I am trying to obtain a probability density function (PDF) from its moment generating function (MGF) and the problem is the nature of the MGF. My MGF is $M(t)=I_o^{\beta}(s)$, where $\beta$ is a positive integer: \begin{equation} M(t)=\left( \frac{1}{\pi}\int_0^{\pi} e^{t\cos \theta} d\theta \right)^{\beta} \end{equation}

I am stuck in finding the PDF from $M(t)$, as this is not a standard form for which a look-up table type approach can be used.

I tried replacing $t$ by $it$ and then taking the inverse Fourier Transform route, but it didn't yield anything. Even if exact solution cannot be found, some sort of approximation or bound will also be helpful.

asked Apr 26 '20 at 07:36

Priyadarshi Mukherjee

Is it in connection with the Von Mises distribution (https://en.wikipedia.org/wiki/Von_Mises%E2%80%93Fisher_distribution) ? – Jean Marie Apr 26 '20 at 09:19
...or non central $\chi^2$ (https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution) ? – Jean Marie Apr 26 '20 at 09:36
No, its actually in connection with the Chebyshev chaotic maps. – Priyadarshi Mukherjee Apr 26 '20 at 10:04
Do we agree that $I_0$ is the modified Bessel function ? – Jean Marie Apr 26 '20 at 10:15
Yes, for sure. I used the integral form of this function here instead of the name "zero-th order modified Bessel function of the first kind" just for the sake of clarity in my question. – Priyadarshi Mukherjee Apr 26 '20 at 10:43
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Connected : https://math.stackexchange.com/q/925545 – Jean Marie Apr 26 '20 at 22:49
Thank you very much @JeanMarie, I will look into the link. – Priyadarshi Mukherjee Apr 27 '20 at 11:19

Density function from moment generating function

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