Linked Questions

2
votes
0answers
34 views

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?
39
votes
4answers
19k views

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 ...
40
votes
3answers
6k views

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 ...
17
votes
5answers
35k views

Expectation of reciprocal of a variable

I am confused in applying expectation in denominator. $E(1/X)=\,?$ can it be $1/E(X)\,$?
14
votes
2answers
11k views

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?
14
votes
7answers
3k views

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 ...
6
votes
2answers
3k views

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 ...
9
votes
2answers
889 views

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 ...
6
votes
2answers
300 views

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 ...
5
votes
1answer
610 views

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 ...
4
votes
2answers
362 views

PMF of the number of trials required for two successive heads

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 ...
3
votes
2answers
123 views

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 ...