# Questions tagged [moment-generating-functions]

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### Finding the moment generating function for independent trial with density $f_X(x)=\frac{e^{|x|}}{2}$

For an independent trial for the random variable X with density $f_X(x)=\frac{e^{|x|}}{2}$. If $S_n = X_1 + ... X_n$, $A_n = S_n/n$, and $S_n^*=\frac{S_n-n\mu}{\sqrt{n\sigma^2}}$, I found the ...
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### Is the moment generating function of the gamma density $g(t)=(\frac{\lambda}{\lambda - t})^n$?

My book defines the gamma density as the following: $$f_X(x)=\lambda (\lambda x)^{n-1}e^{-\lambda x}/(n-1)!$$ And has the moment generating function of this density as $\frac{\lambda}{\lambda +t}$. Is ...
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### 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 $M(T_1 , T_2) = \frac{1}{2} e^{T_1 +T_2} + \frac{1}{4} e^{2T_1 +T2} + \frac{1}{12}e^{T_2} + \frac{1}{6} e^{4T_1 +3T_2}$ ...
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### uniform moment generating function at t=0

I have calculated the moment generating function for the uniform distribution as $$M_X(t)=\frac{e^{tb}-e^{ta}}{t(b-a)}$$ However I know $M_X(0)=1$ but I can't get my head around how this is possible ...
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### Help with integration of first moment from PBE

I'm wondering anyone can help me with the following integration: $$\frac{d(m_0 V)}{dt} = BV$$ where $B$ is just a constant, $V$ is a variable parameter. Product rule must be applied somehow? EDIT: ...
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### The expectation of log[1+e^(f)]

There are many examples about how to compute the expectation of $\log(1+e^x)$ such as approximating it with something like Maclaurin series. I have a slightly complicated situation \begin{equation} \...
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### Moment Generating Functions Taylor series

So I'm revising moment generating functions and I'm stuck on a part of a question I'm looking at. So I am asked to find the moment generating function of a random variable X whose distribution is ...
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### 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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I have the following problem : Given $X \sim N(\mu,\sigma^2)$ and $X' = h(X) = (\frac{x-\mu}{\sigma})^2$ Find $E[X']$ and $V[X']$. My reasoning is as follow : Since $X' \sim (\frac{x-\mu}{\... 1answer 40 views ### Correlation and Moment generating function Let$M=E[e^{t\cdot X}]$be the moment-generating function of the random vector$X$in$\mathbb{R}^n$. Then is it true that $$E[(X -EX)^s] = \left. \frac{\partial \ln{M}}{\partial t^s} \right|_{t=0}$$... 0answers 11 views ### the moment generating function of the negative binomial distribution According to the textbook I use, it states that:$X$~$Neg.bin(x;k,\theta) = {n-1 \choose k-1}\theta^k(1-\theta)^{n-k}$Which I have no problem. The problem arises when I try to find the moment ... 1answer 24 views ### factorial moment generating function I'm trying to get the factorial moment-generating function of a binomial random variable. I know that$F_X(t) = E[t^x] = \Sigma_xt^xp(x)$so I get$\Sigma_xt^x{n \choose x}\theta^x(1-\theta)^1-x$... 1answer 76 views ### Moment generating function of two Poisson distributions The time between accidents on the Riverfront Bridge follows a Poisson process with a mean time of 40 days between accidents. The time between accidents on the Overview Bridge follows a Poisson ... 1answer 36 views ### differentiating(?) Poisson distribution I've been facing this - i don't even know how to call it - problem for a few hours now and I have know idea how to "do" this. I mean... I feel like this has something to do with binomality of Poisson ... 0answers 19 views ### Example of higher random vector moments While reading about random vectors, I learned that... $$E\left[\vec{X}\right] = \left[\begin{array}{cccc} E\left[\vec{X}_1\right] & E\left[\vec{X}_2\right] & \cdots & E\left[\vec{X}_m\... 1answer 456 views ### Need help with moment generating function of geometric distribution The cheat sheet I have tells me the moment generating function for Geometric Distribution is:$$M(t) = \frac{p}{1-(1-p)e^t} $$But most resources and me personally working it out I get:$$M(t) = \... 1answer 178 views ### Poisson Variable with an Exponential Parameter becoming a Geometric Distribution? Suppose Λ ∼ exponential(γ) and X ∼ Poisson(Λ). Use moment generating functions to show that$X + 1 \sim \mathrm{geometric}(p)$and determine$p$in terms of γ. In order to solve this problem, I first ... 0answers 6 views ### Branching processes: Proof a limit of a probability i have tried a question which the last part I couldn’t solve. Previously I have proved the explicit formula of$G_n={[n-(n-1)s]}/{[n+1-ns]}$if that is useful Let Xn be the size of the nth generation ... 0answers 21 views ### How to use mgf to find the distribution of a standardised normal. We are given that:$Z=\frac{Y-\mu}{\sigma}$We want to show that, if$Y\sim N(\mu ,\sigma^2)$, then$Z\$ is a standard normal random variable using the uniqueness of moment generating functions. The ...
For a zero mean sub-Gaussian R.V. we know that: $$\mathbb{E}[e^{\lambda X}]\le e^{\frac{\lambda^2\sigma^2}{2}},\qquad\forall\lambda\in \mathbb{R}$$ From Taylor series expansion and equating the terms ...