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Let $X$ be a random variable with $ E(X^m) = (m+1)! (2^m), \ m=1,2,3, ...$

Find the density of $X$.

I tried finding the nth derivative of the moment generating function to solve this question but somehow I can't find the answer.

werck
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user90596
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1 Answers1

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https://en.wikipedia.org/wiki/Moment-generating_function This Wikipedia article describes the expansion of the moment generation function in terms of the moments (which you have).

herb steinberg
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  • If you use the characteristic function instead of the moment generator, I suspect getting the distribution function (inverse Fourier transform of char. function) would be easier. – herb steinberg Oct 12 '18 at 17:12
  • I don't understand could you solve it? – user90596 Oct 12 '18 at 17:32
  • Char. function $\phi(t)$ can be used to get density function $f_X(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-itx}\phi(t)dt$. – herb steinberg Oct 12 '18 at 20:22
  • $\phi(t)=\sum_{k=0}^{\infty}\frac{E(X^k)}{k!}(it)^k$. It looks very similar to the series for the moment generation function. – herb steinberg Oct 12 '18 at 20:34
  • Ok,I don't know about characteristic functions but I will try to understand . – user90596 Oct 12 '18 at 20:37
  • Characteristic functions are very similar to moment generating functions. They are Fourier transforms of probability density, rather than Laplace transforms (moment generator). The main advantage is that the inverse transform to get the density is usually a lot easier. – herb steinberg Oct 12 '18 at 23:41