# 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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### Proving that mgf determines distribution via Laplace transform

I am reading this question and the answer provided there about the moment generating function (mgf) and how its uniqueness can be proved via the uniqueness of Laplace transforms. In my book, Measure ...
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### what is the formula for calculation of fourth raw moment (or central moment) from variance and 3rd central moment (or raw moment)?

Basically the title. I can't seem to find any solution for this. I have the mean, variance or the second central moment and third central moment and third raw moment. I need to find the fourth raw ...
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### How to find probability from $E[X^n]$?

It is given that $E[X^n] = \frac{2}{5}(-1)^n + \frac{2^{n+1}}{5}+\frac{1}{5}$, where $n=1,2,3,\ldots.$ I need to find $P(|X-\frac{1}{2}| > 1)$. What my approach is : I have opened the modulus ...
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### The distribution function of the min of two random variables which are dependent via a common term

Consider the following two random variables: $$Z_1=U_1-X_1$$ and $$Z_2=U_2-X_1,$$ where $U_1$ and $U_2$ are two i.i.d random variables following a general distribution, and $X_1$ is an exponential ...
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Consider the probability distribution $D \sim X - a$ where $X \sim \text{Exp}(\lambda)$. My task is to find the MGF of probability distribution $D$. I think I have a solution but it contradicts what I ...