# Questions tagged [moment-generating-functions]

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715 questions
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### Distribution of the difference of two normal random variables.

If $U$ and $V$ are independent identically distributed standard normal, what is the distribution of their difference? I will present my answer here. I am hoping to know if I am right or wrong. ...
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### Deriving Moment Generating Function of the Negative Binomial?

My textbook did the derivation for the binomial distribution, but omitted the derivations for the Negative Binomial Distribution. I know it is supposed to be similar to the Geometric, but it is not ...
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### About the “Cantor volume” of the $n$-dimensional unit ball

A simple derivation for the Lebesgue measure of the euclidean unit ball in $\mathbb{R}^n$ follows from computing $$\int_{\mathbb{R}^n}e^{-\|x\|^2}\,dx$$ in two different ways. See, for instance, ...
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### Find the moment generating function of the sum of exponential random variables $S=X_1+X_2+X_3+X_4$

Let $X_1+X_2+X_3+X_4$ be iid exponential random variables with parameter λ, and $S=X_1+X_2+X_3+X_4$ S follows the gamma distribution with parameters $\lambda$ and $r=4$. We know that an exponential ...
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### History of Moment Generating Functions

I am beginning to appreciate how important Moment Generating Functions (MGFs) are regarding various common probability distributions and the ways their expectations/variances are calculated. My open-...
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### Find m.g.f. given $E(X^r)$ function?

"Let $X$ be a random variable with $E(X^r) = 1 / (1 + r)$, where $r = 1, 2, 3,\ldots,n$. Find the series representation for the m.g.f. of $X$, $M(t)$. Sum this series. Identify (name) the probability ...
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### Determine the PDF from the MGF [closed]

If the moment generating function is given as; $\psi_X(s) = e^{s^2}$ How can i determine the PDF of $X$?
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### convexity of log of moment generating function

Why is log of a moment generating function of random variable Z is convex? that is $\log \mathbb{E}[\exp(\lambda.Z)]$ My logic says since expectation is linear so it is in particular convex and ...
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### Showing inequality: $pe^{x(1-p)}+(1-p)e^{-xp} \leq e^{x^2(3/4)p}$ for $0 \leq p \leq 1/2, 0 \leq x \leq 1$?

How can I show that $$pe^{x(1-p)}+(1-p)e^{-xp} \leq e^{x^2(3/4)p}$$ for $0 \leq p \leq 1/2, 0 \leq x \leq 1$? I've been stuck on this for a long time; I tried expanding out the taylor series on ...
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### What is the origin of the term “moments” in the study of random variables?

I understand what the moments are, I just want to know who picked the term "moment" and why? How is the word "moment" related to different but related ways to describe the shape of a random variable?
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### Using the mgf to get moments and the dominated convergence theorem

The moment generating function of $X$ is given by $M_X(t) = E[e^{tX}]$. I'm wondering if it is always possible to obtain moments as $E[X^k] = M_X^{(k)}(0)$, i.e. the $k^\text{th}$ derivative of the ...
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### Show that the moment generating function of $W$ is $M_W(t) = (qe^t+p)^n$

If $Y$ is a random variable with moment-generating function $M_Y(t)$ and if $W$ is given by $W=aY+b$, then the moment generating function of $W$ is $e^{tb}M_Y(at)$ Suppose that $Y$ is a binomial ...
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Let $X_t = \sum_{i=1}^{N_t} Y_i$ and $N_t$ be a Poisson process with intensity $\lambda >0$. Suppose $Y_i$ are i.i.d. (independent of $N_t$) with normal distribution $N(m,\sigma^2)$. Compute $... 3answers 298 views ### Finding moment generating function of$f(x)= \frac 1 {\theta^2} xe^{-x/\theta}$I've been stuck on this question for a while now and my exam is coming up so,any hints/comments etc. would be greatly appreciated. Question: Find the moment generating function of the probability ... 2answers 299 views ### Statistics: Odd Moments Need help with this stat question. I know you start by integrating$z^k f(z)$from$-\infty$to$0 +$integral of$z^k f(z)$from$0$to$\infty$. After that I'm stuck. 2answers 98 views ### Boundedness of an integral of square function implying zero integral Let$\alpha:\mathbb R\mapsto\mathbb R$be the smooth function such that $$\int_{-\infty}^{\infty}[\alpha'(x)-x\alpha(x)]^2e^{-\frac{x^2}2}dx<\infty.$$ I wish to prove that$$\int_{-\infty}^{\infty}[... 1answer 564 views ### Sum of a random number of independent random variables Consider the sum$Y = X_1 + \cdots + X_N$where$N$is a random variable that takes nonnegative integer values, and$X_1, X_2, \cdots$, are identically distributed random variables. Assume that$N, ...
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 $-2\alpha\sum\log x_i$ ...
Let $X$ and $Y$ be independent random variables, with known moment generating functions $M_X(t)$ and $M_Y (t)$ and $I$ be such that $P(I = 1) = 1 − P(I = 0) = p \in (0, 1)$. Compute the moment ...