# 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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### MGF of "generalised" Negative Binomial distribution

Would the "extended" Negative Binomial have the same MGF as Negative Binomial? (See the definition of "extended" Negative Binomial below by Wikipedia) Could someone please help ...
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### Chernoff Bounds for Independent Bernoulli Sums

What is wrong with this proof? Can you notice that? or I am wrong? In my opinion, in the R.H.S. of the inequality (3.2), the index of 'e' is negative but it must be positive if we use the given proof ...
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### Relationship between CDF and MGF?

Suppose $X$ is a random variable $\in (0,1]$ $f(x)$ is the CDF of this random variable $g(t)=\operatorname{mgf}(-t)$ where "mgf" is the moment generating function. Can we infer the ...
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### What distribution has CDF $\propto \psi ^{(1)}\left(x^{-\frac{1}{2}}\right)$

Is there a way to express random variable with the following CDF $g(x)$, $x\ge 0$, in terms of known named distributions? $$g(x)\propto \psi ^{(1)}\left(x^{-\frac{1}{2}}\right)$$ where $\psi^{(k)}$ is ...
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### Existence of moment generating function [closed]

Give a example of discrete random variable for which mgf $M(t)=E\left(e^{tx} \right)$ does not exist . I have tried with geometric(p) distribution when $(1−p)e^t≥1$ , the mgf does not converge. Is it ...
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### Are linear combinations of independent random variables again independent?

Let $X_1,X_2,\ldots,X_n$ be (iid) Random variables and define $Y_n:=\sum_{j=1}^na_jX_j$ with $a_j\in \mathbb{R}$, can we then say that the $a_jX_j$ are independent aswell. Can we express the MGF than ...
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### Double Expectation in Moment Generating Function

Context Assume a game using a fair 6 sided dice ends when one rolls a 6. A random variable N is the total number of throws. N is distributed such that: $p_N(n) = \frac{5}{6}^{n-1}\frac{1}{6}$ Assume ...
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### Generating function of a random walk

Consider a random walk with $S_n=\sum^n_{i=1}X_i$, where the random i.i.d. steps $X_i$ take values $-1,0,2$ with probabilities $1/9,1/9,7/9$ respectively. Set $S_0=1$. I would like to calculate the ...
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### Exam question about compound distributions and MGFs

Studying for a test in course about stochastic processes, here's a test question that I can't fully understand: An insurance company insures its policyholders against damages of a particular kind. ... 100 views

### Calculating the $M_X(t)$ of the pdf $f_X(x) = \frac{\sin(x)}{2}$

I am working on calculating the moment generating function for the pdf $f_X(x) = \frac{\sin(x)}{2}$ with the bounds $[0, \pi]$, and here is my attempt although I would like to know whether I have ...
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### Calculate moment generating function of normal distribution [duplicate]

The moment generating function of a normal distribution is defined as $M(t) = \int_{-\infty}^\infty e^{tx}\frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2} dx$ In a book I’m ...
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### Textbook clarification on Taylor expansion of mgf (Casella, Berger)

In Statistical Inference by Casella, Berger (version 2), Section 2.6.1, we have the following expression for the Taylor expansion of the mgf: $$M_X(t)=\sum_{r=0}^\infty \frac{(-1)^r\mu_r't^r}{r!}$$ ...
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### Moment Generating function of Multinomial Distribution, very confused

I know somewhat similar questions have been asked but they all seem to have somewhat different answers and I'm kind of confused. I am self learning statistics, and I have the following question that ...
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### What is the expectation of $e^X$, where $X$ is a random variable with a geometric distribution?

if $X$ is a random variable with a geometric distribution how can I calculate $$E(e^X)$$ I have no idea on how to do that.
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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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