# Tagged Questions

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### How to prove three properties of the moment generating function? [duplicate]

The moment generating function of a random variable $X$ is defined to be the function $$M_{X}(t)=E(e^{tX})=\sum_{n=0}^{\infty}\frac{E(X^n)}{n!}t^n.$$ Let $I=\{t\in\mathbb R:M_{X}(t)<\infty\}.$ I ...
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### Interpretation of a PDF squared [duplicate]

I have a problem where the crucial variable is the integral of the squared PDF of a random variable, i.e. $\int f(x)^2dx$ How should I interpret this property of a distribution? If $f(x)$ is ...
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### How to show that $\mathrm{mgf}$ $M(s)$ and $\mathrm{pgf}$ $P(s)$ are related?

Let $X$ be an integer-valued $rv$ with $\mathrm{pgf}$ $P(s)$ (probability generating functions) and suppose that $\mathrm{mgf}$ $M(s)$ (moment generating functions) exist for $s∈(-s_0,s_0),s_0>0$. ...
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### Is it possible to calculate mutual information by moments generating functions?

I went to listen to a workshop and some audience asked the presenter how the moments can improve the mutual information. I am learning the MI(Mutual Information) and moments so don't have enough ...
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### Proving that MGF determines PDF when the PDF is defined for whole real line

If two PDFs have the same moment generating function that converges in an open set around 0, then the PDFs are same. This is a well known fact, but I can't find its proof. If the PDFs are defined ...
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### If the n(th) moment exists does it mean all smaller moments exist too?

I would like to prove the following statement: If the $r$th moment of a random variable $X$ exists and is finite, then all moments $1$ to $r-1$ exist and are finite. Edit: I mean the raw ...
### Approximating $Pr[n \leq X \leq m]$ for a discrete distribution
What's the best way to approximate $Pr[n \leq X \leq m]$ for two given integers $m,n$ when you know the mean $\mu$, variance $\sigma^2$, skewness $\gamma_1$ and excess kurtosis $\gamma_2$ of a ...