# Questions tagged [moment-generating-function]

A description of a probability distribution which is related to the Laplace transform. Use also for its logarithm, the cumulant generating function.

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### moment generating function of gamma distribution through log-partition function

How to drive the moment generating function of Gamma distribution using log-partition function? Suppose $X\sim\Gamma(\alpha,\beta)$, gamma distribution with parameter $(\alpha, \beta)$. Then $X$ has ...
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### Moment generating function of non-central Chi-squared distribution with complex mean?

I have random variables $(X_1, \dots, X_k)$ distributed independently according to normal distributions with complex means, i.e. $j\mu_i, i=1\dots k, j^2=-1$, with unit variances. I want to study the ...
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### How to find the MGF of the difference of 2 random variables

Let $X\sim N(12,4)$ and $Y \sim N(3,1)$ Let $Z = X - Y$ Find the Moment Generating Function of $Z$. I tried finding the expected value of $e$ to the power of $tz$, but this isn't possible to separate ...
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### Why is moment generating function (MGF) evaluated at zero?

Although this question touched on MGF exists at neighborhood of 0, I still don't understand why the definition of MGF says $M_X{(t)}$ = E $e^{tX}$ for $t$ in neighborhood of 0. Why must it be 0 but ...
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### Understanding t bounds on MGF

I'm having trouble understanding what the bounds of the $t$ variable are for an mgf. My questions are bolded. Here's an example from a textbook: Suppose X is a random variable for which the pdf is: ...
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