# 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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### Sum of exponential of MVN dimensions

Consider drawing N random variables $x_{i}$ for $1\leq i \leq N$ from a multivariate normal with $\mu = (\mu_1, .. \mu_i, .. \mu_N)$ and $\Sigma_{N \times N} = [\sigma_{ij}]$ (equivalently, N ...
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### If two distributions have the same moments, how different can they be?

Let us suppose we have two distribution functions $F$ and $G$ with shared domain and also shared moments but not necessarily shared moment-generating functions. I have seen from "Whether ...
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### Generating function for Gamma distribution expectation

I encountered this formula in my assignment: $$X\sim \Gamma(\alpha, \beta), 1\le k < \alpha$$ $$E(X^{-k})=\frac{\beta^k}{\prod^k_{i=1}(\alpha-i)}$$ And I wonder what would happen if $k$ is ...
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### What are the moments of the Beckmann distribution?

Let $U=(u_1, u_2)$ and $V=(v_1, v_2)$ be two randomly distributed points on the Euclidean plane assuming bivariate normal distributions $U \sim N(\mu_u, \Sigma_u)$ and $V \sim N(\mu_v, \Sigma_v)$ with ...
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### MGF of the product of a exponential and a bernoulli random variable

Let $𝑍=𝑋𝑌$ , where $X$ and $Y$ are independent, 𝑋 ~𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙(0.01) and 𝑌∼𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖(0.3) Is there a way to find the m.g.f of 𝑍? I know that I can find the C.D.F by doing as ...
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### Determine the joint moment generating function $M_{V,W} (s1, s2)$ of V and W

So the initial question was: Let X and Y be independent random variables with common moment generating function $m_X(s) = m_Y (s) = e^{s^2/2}$. a) Determine the moment generating function $M_V (s)$ of ...
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### Moments of $\text{exp}(-|x|^{1/2})$

I'm supposed to show that all of the moments of the density $\text{exp}(-|x|^{1/2})$ are finite. I'm not convinced this is true though. The $p$th moment is \begin{align*} \mathbb{E}[X^p] &= \int_{-...
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### MGF of sample mean of poisson distribution

Let $X_1,X_2,\dots,X_n\stackrel{iid}{\sim}Poiss(\lambda)$ Let mgf of $X_1$ is given by $M_X(t)=e^{\lambda(e^t-1)}$ and let $\bar{X_n}=\frac{1}{n}(X_1+X_2+\dots+X_n)$ Then, by Weak Law of Large Numbers ...
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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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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 ...