Articles of moment generating functions

Moment generating function of a sum of i.i.d. random variables

Let $\{ Y_j: 1\leq j \leq K \}$ be a collection of i.i.d. random variables. Suppose we have two random variables $W$ and $W’$ that have the same distribution function, where $W’$ is given by: $$W’=\sum_{j=1}^{K} Y_j,$$ where $K$ is a random number. Then: $M_W(s)=G_K(M_Y(s))$, where $M_Y$ is the moment generating function of $Y$. Since […]

The sum of moment generating functions

Let $X, Y$ be independent r.v with moment generating functions $M_X(t)$ and $M_Y(t)$ respectively. Is there a function of $X$ and $Y, Z$, with moment generating function $$\frac{M_x(t) + M_y(t)}2$$

Distribution of $Z$ from Moment Generating Function

Suppose that $X_1, X_2, …, X_n$ are independent and identically distributed Exp(λ) random variables and let $Z = X_1 + X_2 + · · · + X_n$. Determine $M_Z(θ)$, the moment generating function of $Z$ and hence, or otherwise, determine the distribution of $Z$ Working: I know that $M_z(\theta)=\left(\frac{\lambda}{\lambda – \theta}\right)^n$. But I don’t even […]

How to find nth moment?

I’m quite new to the field so please bare with me. Problem: Let ξ be a random variable distributed according to a log-normal distribution with parameters μ and $σ^2$, i.e. log(ξ) is normally distributed with mean μ and variance $σ^2$. Show that the kth moment of ξ is given by $E[ξ^k]=e^{kμ+(k^2σ^2)/2}$ Question: I suppose for […]

Analytical continuation of moment generating function

Let’s say some distribution $F(t)$ has finite moment generating function on an open ball (-R, R). $M(x) = \sum m_n x^n /n!$ Let’s extend $M(x)$ to $M(z)$ on a complex strip $S = \{z| |Re(z)| <R\}$. I need to prove that $M(z)$ is also analytic on $S$(thus it is an analytic continuation). What is the […]

Justifying the Normal Approx to the Binomial Distribution through MGFs

Would absolutely love if someone could help me with this question, in a step by step way to help those who are uninitiated to Statistics and Mathematics. So, I am trying to “prove/justify” through MGFs how as n(the sample size) increases and goes to infinity, a standardized binomial distribution converges to the Standard Normal Distribution. […]

Finding the moment generating function of the product of two standard normal distributions

The following question is on my homework assignment that I cannot figure out: Let U and V be independent random variables, each having a normal distribution with mean zero and variance one. Find the moment generating function of the random variable W = UV . I have looked around online, and cannot find an answer […]

Calculating higher order moments of a product of weigthed average parameters

I am trying to find a closed form or a transformation which simplify the numerical treatment of this multiple integral: $$\int_{x_1=0}^1…\int_{x_N=0}^1 \prod_r x_r^{m_r}\left(\frac {x_r f_r} {\sum_g x_g f_g}\right) ^{n_r} dx_1…dx_N $$ where $m_r$ are nonegative integers, $n_r$ are positive integers, and $f_r\ge0$ constants. There is an answer for moment $m=0$ here. Following the same exponentiation […]

Proof that if $Z$ is standard normal, then $Z^2$ is distributed Chi-Square (1).

Suppose that $Z\sim N(0,1)$ and let $V=Z^2$. Prove that $V\sim \chi^2(1)$. I want to use the method of moment generating functions, because I already understand the proof using the method of distribution functions. I will show my work, and then where I got stuck. Since $Z\sim N(0,1)$, then $\mu=0$ and $\sigma^2=1$, and we have $$M_V(t) […]

Moment generating function of a gamma distribution

If I have a variable $X$ that has a gamma distribution with parameters $s$ and $\lambda$, what is its momment generating function. I know that it is $\int_0^\infty e^{tx}\frac{1}{\Gamma(s)}\lambda^sx^{s-1} e^{-x\lambda}dx$ and the final answer should be $(\frac{\lambda}{\lambda-t})^s$, but how can i compute this? P.S. I know that there are other questions on this site about […]