# Questions tagged [moment-generating-functions]

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250 questions
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### Moment generation function -> characteristic function uniqueness

Here's my proof that moment generation function (if exists) uniquely determines characteristic function. Can you please see how to make it more rigorous or improve in either way (e.g. by citing ...
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### Branching process probability generating function

I'm trying to solve the following exercise but I can't seem to solve it. A branching process $(X_n :n \geq 0)$ has $P(X_0 = 1) = 1$. Let the total number of individuals in the first $n$ generations ...
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### Standard Normal Moments and Combinatorics

At around 16-17 mins in this video, the professor calculates the even moments of the standard normal. If $Z \thicksim N(0,1)$ then $$\mathbb{E}[Z^{2n}] = \frac{(2n)!}{2^n \cdot n!}.$$ The right hand ...
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### Normal Approximation to Binomial Distribution using Moment Generating Functions

We're told not to use the central limit theorem to show that the normal approximation is suitable for a binomial distribution when n tends to infinity. I've managed to show the answer, but it ...
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### Why the probability characteristic function is always exist but moment generation function is not always exist?

I know that the characteristic function is always exist and moment generation function is not always exist instantly but don't know exactly mathematically.
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### How big are the exponential moments of a truncated normal distribution?

Given a random variable $X$ valued on $[-1,1]$ with mean zero. We can use say Hoeffding's Lemma to get $$\mathbb E[e^{\lambda X}] \le e^{\lambda^2/2}$$ I believe this bound cannot be improved much ...
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### Sums of trigonometric functions and polynomials

I have to calculate sums of the following forms $$\sum\limits_{k=1}^nP(k)f_m(kx),$$ where $P\in\mathbb{R}[X]$ and $f_m(x)=\sin^m(x)$ or $f_m(x)=\cos^m(x)$. This problem comes from consideration of ...
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### Moment Generating Function exercises: Knowing that $M_{X}(0)=1$ and $M´_{X}(0)= EX$

Hi guys, Any help with letter b of this exercise from Casella´s Book? I could finish the letter a). But I cant move in letter b). Any help? Thanks
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### Given independent random variables $X_{k}\sim\text{Poisson}(\lambda_{k})$, what is the distribution of $X_{1} + X_{2} + \ldots + X_{n}$?

Show that, if $X_{k}\sim\text{Poisson}(\lambda_{k})$ and they are independent for $1 \leq k \leq n$, then \begin{align*} Y = \displaystyle\sum_{k=1}^{n}X_{k}\sim\text{Poisson}\left(\sum_{k=1}^{n}\...
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### Using MGF's to characterize a distribution

Let $X_1,X_2,X_3$ be independent such that for all $x > 0$, $$P(|X_i| > x) < e^{-x}, \;\;\; i = 1,2,3$$ Prove that if $X_1+X_3$ and $X_2+X_3$ have the same distribution, then so ...
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### Moment generating function of $(W_T, \max W_t)$

Does there exist an explicit formula for the moment generating function $\psi(u, v) = E e^{u W_T + v M_T}$ of the pair $(W_T, M_T)$ where $M_T = \max_{0\leq t\leq T} W_t$? Using the well-known pdf of ...
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### Is there a name for expressions that are invariant under the exchange of raw moments and cumulants?

I'm interested in expressions that are invariant under the exchange of raw moments and cumulants. This is trivially true of all expressions written only in terms of first order moments but nontrivial ...
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### Moment generating function of a piecewise function

I am struggling like crazy with question b. Here is what I have come up with: But if I then test $M_X'(0)$, I don't get the expected value I calculated in part a: Where have I gone wrong?
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### Moment Generating Function of a Beta random variable.

After getting some excellent help on this problem in the statistics SE, I am reformuluating my question. Let me know if I should just delete it and ask a new one. Let $V$ be a $Beta(\alpha,1)$ ...
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### Moment-generating Function of a Continuous R.V. whose P.D.F is 1 from (0, 1)

I have been working on this problem for a few hours now and I feel I am missing something simple. The problem is to find the moment-generating function of a continuous random variable whose ...
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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 \...
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The probability density function of T is given by $$f(t) = 1/2h \text{, for each } t\in(-h,h)$$ where $h > 0$. Derive an expression for the central moment I used integration and got $\frac{(b-u)... 0answers 29 views ### Method of moments when the first moment is$0$I have a quick question regarding the method of moments estimator. Generally, when you have$k$parameters you want to estimate, it suffices to find$k$equations using$k$moments. If you are ... 0answers 42 views ### Calculate Probability from Moment Generating Function Let 𝑋 and 𝑌 be two discrete random variables with the joint moment generating function $$𝑀_{𝑋,𝑌}(t_{1},t_{2})=(\frac{1}{3} e^{t_{1}} + \frac{2}{3})^{2} (\frac{2}{3} e^{t_{2}} + \frac{1}{3})^{3}... 0answers 76 views ### Moment Generating Function - Cauchy Random Variable I want to show that, for every t\neq 0, the moment generating function of the standard Cauchy distribution is equal to +\infty, i.e.$$M_X(t) = \int_{-\infty}^{+\infty} \frac{e^{tx}}{1+x^2}dx = +\... 0answers 24 views ### Transformation of RV's Moments Given a transformation of the RV$X$, how do it's moments transform? More Detailed Formulation Suppose that$X:(\Omega,\mathcal{F},\mathbb{P})\rightarrow \mathbb{R}$, is a random variable whose MGF ... 0answers 36 views ### Establish bound for a probability using moment generating function I have the following question Let$X_{1}$,$X_{2}$, ...,$X_{n}$be independent and identically distributed random variables with moment generating function$M_{X}(t)$, for -h < t < h, ... 0answers 81 views ### Moment generation function - compound Poisson Let$S_1$be distributed with compound Poisson$\lambda_1=2$and discrete indeminizations:$f_1 (x), x \geq 0$. And let$S_2$be compound Poisson distributed with$\lambda_2=4$, and discrete ... 0answers 20 views ### Tight Bounds on Moments of$X = a_1x_1 + \cdots + a_nx_n$Suppose that$x_i,z_i$are i.i.d and that the moments$\mathbb{E}x_i^m \leq K \mathbb{E} z_i^m$uniformly for$m=1,2,\ldots$. For simplicity's sake take$x_i$and$z_i\$ to be symmetric so we can ...
I am stuck with the question below. If X is a negative binomial random variable, then $$Y=r+x$$ is the total number of trails necessary to obtain r S's. Obtain the moment generating function of Y ...