12 questions linked to/from Real life uses of Moment generating functions
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### What does the MGF do when its parameter is other than 0? [duplicate]

I've only seen the MGF used when it takes 0 as it parameter and then derived for the nth moment. What does the MGF tell us when the parameter is other than 0?
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### Generic sum of Gamma random variables

I have read that the sum of Gamma random variables with the same scale parameter is another Gamma random variable. I've also seen the paper by Moschopoulos describing a method for the summation of a ...
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### How does saddlepoint approximation work?

How does saddlepoint approximation work? What sort of problem is it good for? (Feel free to use a particular example or examples by way of illustration) Are there any drawbacks, difficulties, things ...
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### Expectation of reciprocal of a variable

I am confused in applying expectation in denominator. $E(1/X)=\,?$ can it be $1/E(X)\,$?
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### What is the difference between moment generating function and probability generating function?

I am confused between the two terms " probability generating function" and "moment generating function." How do those terms differ?
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### Intuitively understand why the Poisson distribution is the limiting case of the binomial distribution

In "Data Analysis" by D. S. Sivia, there is a derivation of the Poisson distribution, from the binomial distribution. They argue that the Poisson distribution is the limiting case of the binomial ...
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### Expected value of $1/x$ when $x$ follows a Beta distribution

Let $x$ have the probability density: $$f(x) = \frac{x^{\alpha - 1}(1-x)^{1 - \beta}}{\mathrm{B}(\alpha,\beta)}$$ where $\alpha,\beta$ are two positive parameters and $0 \le x \le 1$ is the domain ...
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### When to prefer the moment generating function to the characteristic function?

Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let $X : \Omega \to \mathbb{R}^n$ be a random vector. Let $P_X = X_* P$ be the distribution of $X$, a Borel measure on $\mathbb{R}^n$. The ...
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### For independent RVs $X_1,X_2,X_3$, does $X_1+X_2\stackrel{d}{=}X_1+X_3$ imply $X_2\stackrel{d}{=}X_3$?

Let $X_1,X_2$, and $X_3$ be independent random variables such that $X_1+X_2$ and $X_1+X_3$ have the same distribution. Does it follow that $X_2$ and $X_3$ have the same distribution? Can this be ...
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### Bound for weighted sum of Poisson random variables

Suppose I have some independent Poisson-distributed random variables $X_1 \ldots X_N$ with parameters $\lambda_1 \ldots \lambda_N$. These can be thought of as processes where each arrival/event ...
A coin with probability $p$ of landing a head is tossed repeatedly till the occurrence of two consecutive heads. Let $X$ be the random variable denoting the the number of trials needed. What is the ...
### sum of $N$ gamma distributions with $N$ being a poisson distribution
I have an event having poisson distribution with time intervals of one minute. Every event has accomplishment time with gamma distribution. I $N$ number of events start in $t$ minutes, the what will ...