Questions tagged [moment-generating-functions]

Description added to tag

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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 ...
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2answers
6k views

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 ...
2
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1answer
3k views

Finding the M.G.F of product of two random variables.

We are given two independent standard normal random variables $X$ and $Y$. We need to find out the M.G.F of $XY$. I tried as follows : \begin{align} M_{XY}(t)&=E\left(e^{(XY)t}\right)\\&=\...
2
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1answer
12k views

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^...
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2answers
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Distribution of the difference of two normal random variables.

If $U$ and $V$ are independent identically distributed standard normal, what is the distribution of their difference? I will present my answer here. I am hoping to know if I am right or wrong. ...
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1answer
974 views

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 ...
5
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2answers
359 views

Asymptotic Moments of the Binomial Distribution, $E(X/(np))^k = 1 + O(k^2/n)$?

Let $X \sim \text{Binomial}(n, p)$ be the sum of $n$ Bernoulli($p$) random variables. What is the value of $E(X/(np))^k$, where $k$ is a large integer, as $n$ grows large? From calculations the ...
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1answer
167 views

Linear combination of power law distributions

I am exploring whether a linear combination of power law distributions is also a power law distribution. Specifically, if $X \sim (\alpha -1) x_{\min}^{(\alpha -1)} x^{-\alpha}$ and $Y \sim (\beta -1) ...
1
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1answer
311 views

Strong Law of Numbers for $S_{n}$ Bounded Casella Berger 5.38

So this is from Casella Berger 5.38 b) the question states `` Let $X_{1},...,X_{n}$ be iid with mgf $M_{X}(t)$. Let $S_{n} = \sum X_{i}$ and $\bar{X_{n}}= \frac{S_{n}}{n}$. use the fact that $M_{X}...
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1answer
28 views

Find the Distribution that corresponds to the given MGF

I am working on a problem and am a little bit confused. I need to find the distribution that corresponds to the MGF: $2e^t\over3-e^t$ Do we need to separate this into something like: 2e$^t$ and $1\...
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1answer
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Tail bounds for maximum of sub-Gaussian random variables

I have a question similar to this one, but am considering sub-Guassian random variables instead of Gaussian. Let $X_1,\ldots,X_n$ be centered $1$-sub-Gaussian random variables (i.e. $\mathbb{E} e^{\...
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2answers
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Deriving Moment Generating Function of the Negative Binomial?

My textbook did the derivation for the binomial distribution, but omitted the derivations for the Negative Binomial Distribution. I know it is supposed to be similar to the Geometric, but it is not ...
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3answers
7k views

$X$ standard normal distribution, $E[X^k]=?$

I'm stuck with a homework problem where we are supposed to prove that the expected value $E[X^k]$, if $X$ has standard normal distribution, is equal to: $$E[X^{2k}]=\frac{(2k)!}{k!\cdot2^k}.$$ But I ...
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1answer
2k views

Derivative of moment generating function

If the moment generating function of $X$ exists, i.e., $$M_X(t)=E[e^{tX}],$$ then the derivative with respect to $t$ is usually taken as $$\frac{dM_X(t)}{dt}=E[Xe^{tX}].$$ Usually, if we want to ...
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2answers
511 views

finite variance but infinite higher moments

Is it possible to find a positive random variable with finite variance such as \begin{equation*} \mathbb{E}(X^{2+\varepsilon})=+\infty \end{equation*} for all $\varepsilon > 0$ ? Equivalently, is ...
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1answer
437 views

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 $S = \{z| |Re(z)| <R\}$...
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2answers
3k views

The derivatives of the logarithm of a moment generating function

Let $M_{X}(t)$ be an mgf of $X$. Show that the first derivative of $\ln M_{X}(t)$ at $t=0$ is $\mathbb{E}[X]$ and the second derivative of $\ln M_{X}(t)$ at $t=0$ is $\text{Var}[X]$ I'm not ...
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2answers
6k views

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 ...
2
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1answer
107 views

Computing $\iint_{\mathbb R^2}\exp\left(u\frac{xy}{\sqrt{x^2+y^2}}+v\frac{x^2-y^2}{2\sqrt{x^2+y^2}}-\frac12(x^2+y^2)\right)dxdy$

I'm trying to work on another solution to this question by computing the moment generating function: $$M(u,v)=\iint_{\mathbb{R}^2}\frac1{2\pi}\exp\left(u\frac{xy}{\sqrt{x^2+y^2}}+v\frac{x^2-y^2}{2\...
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0answers
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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 ...
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1answer
2k views

Moment generating function of sample mean and limiting distribution

Problem: Let $\bar{X_{n}}$ be the mean of a random sample of size n from an exponential distribution with the following density fctn: $$f(x) = e^{-x}, 0<x<\infty \space, 0 \space otherwise$$...
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2answers
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Expected value of the Max of three exponential random variables

So the question asks: Let $X_1,X_2,X_3\sim \operatorname{Exp}(\lambda)$ be independent (exponential) random variables (with $\lambda> 0$). (a) Find the probability density function of the ...
3
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1answer
605 views

Characteristic function and moment generating function: differentiating under the integral

In order to justify the interchange of the derivative and integral when differentiating a characteristic function, one can use the dominated convergence theorem: $$\frac{d}{dt} \int e^{itx} P(dx) = ...
2
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1answer
964 views

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'=\...
2
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2answers
138 views

Show that as $d$ goes to $\infty$, a standardized version of $X$ has the STD Normal Dist

I am currently stuck on this problem and I would greatly appreciate some help. The problem is as follows: Let $X$ have a chi-square with $d$ degrees of freedom. Show that a standardized version of $X$...
2
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1answer
906 views

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 $$\frac{M_x(t) + M_y(t)}2$$
2
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3answers
143 views

Let $X$ and $Y$ be independent and identically distributed random variables with moment generating function then $E(\dfrac{e^{tX}}{e^{tY}})$

Let $X$ and $Y$ be independent and identically distributed random variables with moment generating function $M(t)=E(e^{tX});\ \ -\infty<t<\infty$ then $E(\dfrac{e^{tX}}{e^{tY}})$ equals ? $(A)=...
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1answer
81 views

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 $Z$ ...
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2answers
574 views

Given a joint moment generating function, find $P(X<Y)$

Let random variables $X$ and $Y$ have joint MGF: $$M(t_1,t_2) = 1/2e^{t_1+t_2} + 1/4e^{2t_1+t_2} + 1/12e^{t_2} + 1/6e^{4t_1+3t_2}$$ I now need to find $P(X<Y)$. I know how to find the moments as I ...
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2answers
168 views

Bounding the expectation of a function of a zero-mean random variable

I have a random variable $X$ with mean zero, $E[X]=0$, and finite second moment, $E[X^2]=\sigma^2<\infty$. I'm wondering if it's possible to show the following bound: $$ E[(e^{X/2}-1)^2] \leq c\...
1
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2answers
239 views

Expected value Taylor series

If $X ∼ N(0, σ^2)$, use the moment generating function of X to show that $E(X^r)= 0$ for r odd, and $E(X^{2r} ) = \frac{(2r)!σ^{2r}}{2^r(r!)}$ for $r=0,1,2,3.....$ Attempt: $m_X(u) = E(e^{Xu}) = E(...
1
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1answer
89 views

Show $\lim_{p\to 0} m_Y(u)=\Big(\frac{1}{1-2u}\Big)^r$ where $Y=2pX$

For any $0<p<1$ and $r$ a positive integer, the probability function $$f(x)={{r+x-1}\choose{x}}p^r(1-p)^x \ \ \ \ \ \ x=0,1,2...$$ defines a random variable $X$. I have computed the mgf of ...
0
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1answer
6k views

Calculating factorial moment for distributions

I don't really understand how to calculate factorial moment for distributions besides just looking at the given formulas in my textbook. So say I want to calculate the E[X(X-1)] factorial moment for ...
0
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1answer
1k views

MGF of The Negative Binomial Distribution

For any $0<p<1$ and $r$ a positive integer, the probability function $$f(x)={{r+x-1}\choose{x}}p^r(1-p)^x \ \ \ \ \ \ x=0,1,2...$$ defines a random variable $X$. Compute the mgf of $X$ to ...
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1answer
79 views

Find pdf of sum of n indp exp RVs w/o using MGFs

From Williams' Probability w/ Martingales: Re $E[f(S_n)]$, how do I obtain $f_{S_n}(s)$? It seems that $$f_{S_n}(s) = \frac{s^{n-1} e^{-\lambda s} \lambda^n}{(n-1)!}$$ I tried computing $M_{S_n}(t) ...
0
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1answer
41 views

Moment generating function of i.i.d

I was reading this pdf https://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2003/lecture-notes/lec15.pdf I have two questions I know about moment generating function of ...
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0answers
682 views

Finding the moment generating function for an absolute normal distribution

Another exam review question. Suppose I know $Z \sim \text{N}(0,1)$, $Y=|Z|$. And I want to find the moment generating function $M_{Y}(t)$. In our notes it's just given to us as $2 N(t) e^{\frac{t^{...
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2answers
968 views

Finding Variance from a joint moment generating function

The random vars X and Y have, for all real values of $T_1, T_2$, the joint mgf $M(T_1 , T_2) = \frac{1}{2} e^{T_1 +T_2} + \frac{1}{4} e^{2T_1 +T2} + \frac{1}{12}e^{T_2} + \frac{1}{6} e^{4T_1 +3T_2}$ ...
0
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1answer
479 views

Moment generating function (MGF) of the ratio distribution $\displaystyle\frac{X}{Y}$

If we know the moment generating functions (MGFs) of the random variables $X$ and $Y$ to be $M_{X}(s)$ and $M_{Y}(s)$, respectively. The MGF of the sum $X+Y$ will $M_{X}(s) \cdot M_{Y}(s)$. So what ...
0
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1answer
326 views

MGF for Stopping Time

Let $Y_{i}$ be IID with the property $P(Y_{i}=1)=\frac{1}{2}$ and $P(Y_{i}=-1)=\frac{1}{2}$ then if $T=\min\{n:K_n=-c$ or $K_n =c\}$ with $K_n=\sum_{i=1}^{n}Y_{i}$, is a stopping time. Find the Moment ...
0
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2answers
78 views

Central moments versus moments of a normal random variable

Is this statement correct: $\mathbb{E}[(X_1-\mu)^4]$ of $X_1 \sim N(\mu,\sigma^2) $ is the same as $\mathbb{E}[X_2^4]$ of $X_2 \sim N(0,\sigma^2)$? If so is there an easy way to show this? I used ...