Questions tagged [moment-generating-functions]

712 questions
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Let X ∼ Expo(λ). You can assume you know that λX ∼ Expo(1), and that the nth moment of an Expo(1) random variable is n!. Find the skewness of X.

This is a question for class. My prof has given me some guidance but I can't wrap my mind around it. My prof said I should start by considering the equation for skewness very generically, and use the ...
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$\frac{p(1-p)e^t}{(1-p+p e^t)^2} \le \frac 1 4$: Use probability to prove or interpret.

While we can prove the following with AM-GM inequality, $$\frac{p(1-p)e^t}{(1-p+p e^t)^2} \le \frac 1 4, t \in \mathbb R, p \in [0,1]$$ I want to find a proof that involves probability or ...
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how to show that $\frac{\theta e^t(1-\theta)}{(1-\theta+\theta e^t)^2} \leq\frac{1}{4}$?

let $0 \leq \theta \leq 1$ , then how to show that $\forall t\in R$ $$\frac{\theta e^t(1-\theta)}{(1-\theta+\theta e^t)^2} \leq\frac{1}{4}?$$ This is a step of a proof of hoffeding's lemma.
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Application of approximation of moments on poisson distribution

Exercises 101 in Chapter 4 Of the book "Mathematical Statistics and Data Analysis" by Rica states: Find the approximate mean and variance of Y = √ X, where X is a random variable following a ...
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Approximation method of moments (delta) method - worked out example

I am self-studying the delta method, but i cannot understand it. Here i provide an worked out example from Rice in hos book "Mathematical Statistics and Data Analysis". How are the calculations ...
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MGF for Binomial Distribution

I’m learning towards an exam I have. In one of the questions I've being asked to compute the MGF for the binomial distribution. My answer is slightly different from the official answer published by ...
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MGF of squared of inverse gaussian (IG) random variable

If x is an inverse Gaussian random variable, $x\sim IG(\lambda,\mu)$, i.e it is distributed by $f_x(x)={\sqrt{\lambda}\over\sqrt{2\pi x^3}} \exp(-\frac{\lambda(x-\mu)^2}{2\mu^2x}), \quad x>0$ ...
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PDF for Moment Generating Function $(1 +\beta t)^{-\alpha}$ when $\beta > 0$

I'm trying to find the PDF for the following MGF. $$(1 + t/2)^{-3/2}$$ I already know that $(1 - \beta t)^{-\alpha}$ when $\beta > 0$ is an MGF for the Gamma distribution $\Gamma(\alpha, \beta$). ...
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Using MGF's to characterize a distribution

Let $X_1,X_2,X_3$ be independent such that for all $x > 0$, $$P(|X_i| > x) < e^{-x}, \;\;\; i = 1,2,3$$ Prove that if $X_1+X_3$ and $X_2+X_3$ have the same distribution, then so ...
I am given two variables $Y_1$ and $Y_2$ obeying an exponential distribution with mean $\beta= 1$ We are asked what the distribution of their average is and the solution must be found using moment ...