Questions tagged [moment-generating-functions]

712 questions
92 views

33 views

208 views

Log of the Moment Generating Function

The Question: Let $X$ be a random variable such that $\Bbb E[X] = 0$ and $\text{Var}(X)=1$. Show that, as $t \rightarrow 0$, $$\ln M_X(t)= c t^2 + o(t^2)$$ where $M_X$ is the MGF of $X$, $o$ is the ...
194 views

Intuition behind moments of random variables [duplicate]

I am looking to understand the intuition behind moments of random variables. I understand the first moment relating to the mean and the second one relating to the variance. But what use does the n-th ...
25 views

plugging binomial moment function into poisson moment function

I have no idea how the author got from $$E[(pe^s+(1-p)e^t)^x]\rightarrow e^{\lambda(pe^s+(1-p)e^t-1)}$$ The author writes $a=e^t$, and then sets $a=pe^s+(1-p)e^t$ and plugs it in where $e^t$ ...
75 views

conditional moment generating function author transforms solution

At the last step of the provided solution to this problem the author transforms $$\frac{e^{t(n+1)}-1}{e^t-1}=(1+e^t+e^{2t}+...+e^{nt})$$ It is probably something simple, but I am having trouble ...
107 views

Derivation of MGF of a squared random variable X~N(0, 1)

Consider a continuous random variable $Y = X^2$ such that $X\sim N(0, 1)$. The PDF of this variable is $f_Y(x)= \dfrac{1}{\sqrt{2\pi}} x^{-\frac12}e^{-\frac{x}{2}}$ as discussed in this post. ...
36 views

Alternating series of normal random variables and moment generating functions

Say $X_m$ is a normal distribution with mean $0$ and variance $2m$. Now suppose we have an alternating series $-X_1+X_2-X_3+...$ up to $X_n$. Find the distribution of this series. My working so far: ...
32 views

238 views

47 views

Distribution of Product of Arbitrary Number of Random Variables

Suppose $X_i\sim f_{X_i}(x)$, with each $X_i$ independent and identically distributed for some arbitrary probability density function $f_{X_i}(x)$. Given this information, and this information only, ...
91 views

Unique solution to an integral equation? (Moment generating function)

I have a solution to the following integral equation: \begin{equation} \gamma(t) = \dfrac{2}{t}\int_{0}^{1} \gamma(t x)(e^{(1-x)t}-1)dx, \, \gamma(0)=1. \end{equation} Can one show that the ...