# Questions tagged [moment-generating-functions]

This tag is for questions relating to moment-generating-functions (m.g.f.), which are a way to find moments like the mean$~(μ)~$ and the variance$~(σ^2)~$. Finding an m.g.f. for a discrete random variable involves summation; for continuous random variables, calculus is used.

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### Does proximity of moment generating functions implies proximity of characteristic functions?

Let's assume that $U$ and $V$ are non-negative random variables. Suppose that \begin{align} \sup_{t \ge 0 } \frac{| M_U(-t) - M_V(-t)|}{t} \le \epsilon \end{align} where $M_U(t)$ and $M_V(t)$ ...
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### Determining moment generating function $\sum_{i=1}^n iX_i$

Let $X_i \sim Ber(0.5)$ and $X_i$'s independent. Let $Y$ be a random variable with the same distribution as $\sum_{i=1}^n iX_i$. Determine the moment generating function of $Y$. I figured the moment ...
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### Moment generating function doubts

It´s a continous random variable. I have to get the MGF from a piecewise density function, but then, when I have to get the $\mathbb (x)$, the result is undefined, so I don't know if it's correct or I ...
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### Showing that the $r^{th}$ derivative of the moment generating function is the $r^{th}$ raw moment

Is this a sound way of defining the $r^{th}$ derivative for the moment generating function $M_{X}(t)$—in order to show that when evaluated at $0$ it gives us the $r^{th}$ raw moment $E[X^{r}]$ ? ...
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### Possible and Impossible Moment-generating function

$$M_1(t)=0.5(e^t + e^{-t}) \quad t\in R$$ $$M_2(t)=0.5(e^t - e^{-t}) \quad t\in R$$ $$M_3(t)=2*e^t - e^{2t} \quad t\in R$$ It's easy to see that all of the moments of M1 are exists, because all of ...
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### Confusion about expectation / moment generating function / distributation

(I am currently studying a high dimensional probability course with very little background knowledge in probability theory as a whole, so I hope it's not annoying that I seem to be oblivious about ...
In the answer for the question Construction of a MGF for conditional random variables the MGF is given: $M(\theta)= \left(\sum_{i=1}^n p_ie^{\theta a_i}\right)^A$, where $a_i \in R, i=1, \ldots, n$, $... 2answers 19 views ### Differences Between Characteristic Function and Moment-Generating Function Why can't the moment generating function be defined for all random variables, while the characteristic function can be defined for all random variables? 1answer 14 views ### MGF Bernouli random variable Question: If the sequence of moments of a random variable is given as$m_k = \frac{1}{2}(C^k+ (-1)^kC^k), c\in \mathcal{R}$, find the corresponding distribution. The given answer is that the random ... 0answers 57 views ### A bound on the moment generating function Let$X$be a random variable such that$|X|\le{K}$for some$K>0$. I am trying to prove the following bound on the moment generating function of X: $$\mathbb{E}[\exp(\lambda{}X)]\le{}\exp(g(\... 2answers 67 views ### Moments do not characterize the distribution function Let X \sim N(0,1) and Y = e^X. Consider another random variable Z and its PDF is$$f_Z(t) = [1 + \sin(2\pi \log_e(t))] f_Y (t)\quad 1\{t \ge 0\}\quad (1)$$a) Show that (1) is a valid PDF, ... 1answer 36 views ### Exploring Expectation Identity Let X\sim N(\mu ,\sigma^2) and g : \Bbb R \to \Bbb R be a differentiable function and E[|g'(X)|] < \infty. a) Show that E[g'(X)] = \sigma^{-2}E[g(X)(X-\mu)] b) Make use of the above ... 1answer 25 views ### MGF of a probability function [closed] The MGF of p(Y=y) = Σ(from 0 to y) c(n,x) * p^x * (1-p)^(n-x) + c(n-y,y-x) * p(y-x) * (1-p)^(n-x). 1answer 26 views ### Determine the distribution of a function of variable through expectation of power functions Let X be a random variable (r.v.) defined on D\subset{\mathbb{R}}. Consider Y(X) as a function of X, which can be treated as another r.v. Now suppose$$\mathrm{E}(Y^k) = \int_D Y^k(X)dF(x),$$... 1answer 20 views ### Find the moment generating function of r_n Y_n Let (Y_n)_{n\geq 1} be a sequence of geometrical random variables with parameter r_n such that r_n\to 0 as n\to \infty. Find the moment generating function of r_n Y_n. You can ... 2answers 84 views ### Determining a random variable through the Taylor expansion of its moment generating function Let X be a random variable defined on a compact set K\subset \mathbb{R}. The moment generating function (MGF) of X, denoted as M_X(t), t\in \mathbb{R}, is defined as$$M_X(t) = \mathrm{E} [e^{... 0answers 27 views ### What strategy to use to compute this limit? In a statistics problem, the following limit (on the left side) falls out from using various theorems on moment generating functions. The last step of the problem is to show that the limit on the left ... 0answers 36 views ### Density function from moment generating function I am trying to obtain a probability density function (PDF) from its moment generating function (MGF) and the problem is the nature of the MGF. My MGF is$M(t)=I_o^{\beta}(s)$, where$\beta$is a ... 0answers 18 views ### module of characteristic function <1 I have problems with this proof: How to prove that the module of characteristic function is less than one. Why$\vert\mathbb{E}[𝑒^{𝑖𝑡𝑋}]\vert\le \mathbb{E}[\vert𝑒^{𝑖𝑡𝑋}\vert]$? 0answers 23 views ### Expectation and moments inequality X is a standard normal random variable. I came across this inequality in a paper -->$ | E[X^2] | \leq \frac{1}{2} E |X|^3$? I am not sure I understand how this works, can someone explain? 1answer 22 views ### Continuous Analogues of Generating functions A generating function is a useful encoding of a sequence$\{a_n\}_{n=1}^\infty$. That is, given a generating function$A(x)$we can find any$a_n$by taking derivatives. Is there a continuous analogue ... 2answers 32 views ### Expectation of$E[e^{-X^2}]$of a Gaussian distribution Given Gaussian distribution$X \sim N(0, t)$, what is$E\left[e^{-X^2}\right]$? I used the Taylor expansion and Moment Generating Function to get this far:$E(e^{-X^2}) = 1 - E(X^2) + \frac{E(X^4)}{...
I'm trying to improve my understanding on inhomogeneous Poisson processes. In particular, I'm focusing on stochastic integrals of the form $$I(T)=\int_0^Tf(t)dW_t$$ with $W_t$ arrival times of a ...