# Questions tagged [moment-generating-functions]

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### Prove the Random Sample is Chi Square Distribution with Moment Generating Function.

$X_1$, $X_2$, and $X_3$ are random sample taken from normal distribution with $\mu=0$ and $\sigma^2=1$ (standard normal distribution). Let $Y=X_1^2+X_2^2+X_3^2$. Prove that $Y$ have distribution chi ...
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### Opposite of 'If X,Y are independent, Mx+y(t)=Mx(t)My(t)'

It is known that If X,Y are independent random variables, Mx+y(t)=Mx(t)My(t) It's because if X and Y are independent, f(X) and g(Y) are also independent. How about the opposite? It X and Y have ...
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### Use the MGF to derive all moments of $X \sim N(0, \sigma)^2$

Use the moment generating function to obtain the moments of all orders of $X$ if $X \sim N(0, \sigma^2)$ My attempt: The mgf of a generic $N(\mu, \sigma)^2$ is $M_X(t) = e^{\mu t}e^{(\sigma^2t^2)/2}$...
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### Distribution of Dot-Product of Two Independent Multivariate Gaussian Vectors

Let $X,Y\stackrel{\text{i.i.d.}}{\sim}\mathcal{N}(0,I_d)$, where $I_d$ is the $d$-dimensional identity matrix. What is the distribution of $\langle X,Y\rangle=X^TY$? Approach 1: So far I know that ...
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### Sums of trigonometric functions and polynomials

I have to calculate sums of the following forms $$\sum\limits_{k=1}^nP(k)f_m(kx),$$ where $P\in\mathbb{R}[X]$ and $f_m(x)=\sin^m(x)$ or $f_m(x)=\cos^m(x)$. This problem comes from consideration of ...
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### Proof of Simple Facts about Moment Generating Functions

This is surely a very simple, well-known fact about moment-generating functions, though I am interested in the rigor required to prove it. It is surely the case that for any random variable, $X$, ...
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### Why use the exponential in moment-generating function?

When trying to understand the moment-generating function, I've stumbled upon this general function: Mx(t) = E[e^tx], t ∈ R I understand that you have X = (X1, ... Xn) a series of random variables ...
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### Finite Moments of Vector in Exponential Family

I am studying some notes on exponential families and there is a section on the computation of moments. The exponential family has the form $$\exp(\sum_{j = 1}^k \phi_j B_j(x) + C(x) - D(\phi))$$ I ...
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### Integration by Substitution in $\int_0^{\infty}x^r\frac 1{\sqrt{2\pi}x}e^{-(\log x)^2/2}[\sin(2\pi\log x)]dx$
In Casella and Berger (2002) I found an example for non-unique moments (example 2.3.10 on page 64). They are providing the following 2 pdfs: $f_1(x) = \frac 1{\sqrt{2\pi}x}e^{-(\log x)^2/2}$, where \$...