# Questions tagged [moment-generating-functions]

716 questions
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### A proof regarding the moment generating function.

We are required to prove that two random variables $(X,Y)$ are independent if and only if $m_{X,Y}(t_1,t_2)=m_X(t_1)m_Y(t_2)$ where $m(\_)$ is the moment generating function. Supposing $(X,Y)$ are ...
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### Showing for a variable $X$ with mean and variance 1 that $\mathbb{E}\left(e^{X}\right)\geq e$

Let $X$ be a discrete non-negative variable with $\mathbb{E}\left(X\right)=Var\left(X\right)=1$. I'd like to show that necessarily $\mathbb{E}\left(e^{X}\right)\geq e$. My initial intuition was to ...
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### Stochastic Geometry : Obtaining an Integral

I was reading some paper on stochastic geometry when i came across this integral. I was trying to verify results myself but this one seems to be disappointing. Although i do the instructions ...
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### Mathematical Statistics - Jun Shao - 2.12

I'm trying to solve the following problem: Show that the m.g.f. of the gamma distribution $\Gamma(\alpha,\gamma)$ is $(1-\gamma t)^{-\alpha}$, $t<\gamma^{-1}$, using Theorem 2.1(ii). This ...
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### Confidence Interval for Chi-squared distribution

Let $X_1,\ldots,X_n$ be a random sample of size $n$ from a Beta distribution with parameters $α$ and $β=1,$ that the pdf is given by $f(x) = αx^{α-1}$ Find the distribution of $-2\alpha\sum\log x_i$ ...