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A description of a probability distribution which is related to the Laplace transform. Use also for its logarithm, the cumulant generating function.

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Non-central t-distribution, mgf. What is the moment generating function of non-central t-distribution?
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I've recently come across the concept of combinants while reading about probability theory. The Wikipedia article on combinants provides a basic overview but doesn't go into much any detail about how ...
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As I understand it, the Beta Distribution is uniquely defined in terms of its moments (i.e. the Moment Problem has a unique solution on the values of its moments). The Wikipedia article of the Beta ...
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Consider a continuous random variable $X\equiv\log(Y)$. Assume that $$ E(\exp(\alpha X))< \infty \quad \text{ for some $\alpha>0$} $$ I would like to understand what does this assumption imply ...
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I am reading this question and the answer provided there about the moment generating function (mgf) and how its uniqueness can be proved via the uniqueness of Laplace transforms. In my book, Measure ...
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Basically the title. I can't seem to find any solution for this. I have the mean, variance or the second central moment and third central moment and third raw moment. I need to find the fourth raw ...
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It is given that $E[X^n] = \frac{2}{5}(-1)^n + \frac{2^{n+1}}{5}+\frac{1}{5}$, where $n=1,2,3,\ldots.$ I need to find $P(|X-\frac{1}{2}| > 1)$. What my approach is : I have opened the modulus ...
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I have a random variable $X$ with moment generating function: $$m_X(t) = \frac{2}{9} + \frac{e^{-t}}{9} + \frac{e^{-2t}}{9} + \frac{2e^{t}}{9} + \frac{e^{2t}}{3}.$$ I want to find the probability $\...
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I was thinking of a hypothetical distribution where the mean(first cumulant) is non-zero, second cumulant(variance) is zero, and the third cumulant(skewness) is non-zero. The higher order cumulants ...
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Suppose we have $X\sim \textrm{Poisson}(\lambda)$ and we know that moment generating function $M(t)=\mathbb{E}(e^{tX})$. How do we use the moment generating function property $M^k(0)=\mathbb{E}(X^k)$ ...
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I have an array of values of MGF (it is evaluated at some points). The plot of it is shown (blue curve): . Is it possible to find PDF knowing MGF in such form? I tried to fit MGF with some curve (you ...
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Would the "extended" Negative Binomial have the same MGF as Negative Binomial? (See the definition of "extended" Negative Binomial below by Wikipedia) Could someone please help ...
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As the title suggests, I would like to find the MGF of the max of iid exponential random variables. Assume $Z=\max(x_{1},...,x_{n})$, where $x_{i}$ is distributed as exponential($\beta$) and has pdf $\...
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I'll state what I'm trying to prove below. For a Poisson process $N(t) \sim \operatorname{Poisson}(\lambda t)$, $$ P\left(S_n \leq t\right)=P\left(N(t) \geq n\right)=1-P(N(t)<n), $$ where $S_n=\...
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What is wrong with this proof? Can you notice that? or I am wrong? In my opinion, in the R.H.S. of the inequality (3.2), the index of 'e' is negative but it must be positive if we use the given proof ...
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Suppose $X$ is a random variable $\in (0,1]$ $f(x)$ is the CDF of this random variable $g(t)=\operatorname{mgf}(-t)$ where "mgf" is the moment generating function. Can we infer the ...
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Is there a way to express random variable with the following CDF $g(x)$, $x\ge 0$, in terms of known named distributions? $$g(x)\propto \psi ^{(1)}\left(x^{-\frac{1}{2}}\right)$$ where $\psi^{(k)}$ is ...
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Give a example of discrete random variable for which mgf $M(t)=E\left(e^{tx} \right)$ does not exist . I have tried with geometric(p) distribution when $(1−p)e^t≥1$ , the mgf does not converge. Is it ...
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Let $X_1,X_2,\ldots,X_n$ be (iid) Random variables and define $Y_n:=\sum_{j=1}^na_jX_j$ with $a_j\in \mathbb{R}$, can we then say that the $a_jX_j$ are independent aswell. Can we express the MGF than ...
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Context Assume a game using a fair 6 sided dice ends when one rolls a 6. A random variable N is the total number of throws. N is distributed such that: $p_N(n) = \frac{5}{6}^{n-1}\frac{1}{6}$ Assume ...
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Let $X$ be a non-negative random variable with finite variance. It is obvious that its MGF $E[e^{-\lambda(X-E[X])}]$ exists for $\lambda > 0$. How to prove that $E[e^{-\lambda(X-E[X])}] \le \exp(\...
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Consider the following two random variables: $$Z_1=U_1-X_1$$ and $$Z_2=U_2-X_1,$$ where $U_1$ and $U_2$ are two i.i.d random variables following a general distribution, and $X_1$ is an exponential ...
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Consider the probability distribution $D \sim X - a$ where $X \sim \text{Exp}(\lambda)$. My task is to find the MGF of probability distribution $D$. I think I have a solution but it contradicts what I ...
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Suppose that the discrete random variable $X_{n}$has a geometric distribution given by $$f_{X_n}(x_n)=P_n{(1-P_n)}^{x_n}$$ where $$x_n={0,1,2,3,}$$ and $$P_n\ =\ \frac{\lambda}{n}$$ for $0<\lambda&...
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Let's suppose $$E\left(\frac{1}{x}\right)=\int_{0}^{\infty}{M_X(-t)dt}$$ Could you please help me to find $$E\left(\frac{1}{x}\right)$$ where $$X \sim\textrm{Gamma}(\alpha, \beta)$$ and $$E(X) = \...
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I am stuck on the first part of problem 8.2 of the book "A Probabilistic Theory of Pattern Recognition" by Luc Devroye: Show that for any $s > 0$, and any random variable $X$ with $\...
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Could you please help me to prove the following equation: $$E(x^{-1})=\int_{0}^{\infty}M_{x}(-t)dt$$ Where $M_{x}(-t)$ is the moment-generating function. I think the following equation will be useful: ...
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I am stuck trying to solve the following exercise.. Let $X: \Omega\to [a,b] \subset \mathbb R$ be a uniformly distributed random variable. Compute the n-th moment of $X$, i.e. compute $\mathbb E[X^n]$ ...
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Suppose we have independent family of random variables $\{Y\}_{i\in\mathbb{N}}\cup\{N\}$, where $Y$s are identically distributed. Next consider a sum of random number of random variables $W_N\equiv\...
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I have a simple question about moment generating functions(MGFs). Does the interval on which a MGF is defined corresponds to the support of the random variable? For instance, considering a standard ...
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I am studying the Moment Generating Functions of discrete random variables and I got an exercise asking to derive the mgf of a Bernoulli variable and its expected value. I start from the definition of ...
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Let $X$ be a continuous random variable with PDF \begin{equation} \nonumber f_X(x) = \left\{ \begin{array}{l l} 4x^3 & \quad 0 < x \leq 1\\ 0 & \quad \text{otherwise} \end{array} \right. \...
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I am trying to find variance of Y=1/(1+abs(x)) where X is Gaussian RV with mean m, var sigma^2To do that my initial step is to find MGF. can anyone give me ...
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$X$ is a discrete random variable from power series family (e.g., binomial, poisson etc.). is it possible to find an upper bound for the m.g.f of $X$? N.B: from stack exchange I obtained the following ...
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Let $Y$ be a random variable which is a function of another random variable $X$, such that $Y=aX$, where $a$ is a constant. Is it possible that the moment generating function (MGF) of $Y$, is given by,...
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Consider drawing N random variables $x_{i}$ for $1\leq i \leq N$ from a multivariate normal with $\mu = (\mu_1, .. \mu_i, .. \mu_N)$ and $ \Sigma_{N \times N} = [\sigma_{ij}]$ (equivalently, N ...
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Let us suppose we have two distribution functions $F$ and $G$ with shared domain and also shared moments but not necessarily shared moment-generating functions. I have seen from "Whether ...
Galen's user avatar
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I encountered this formula in my assignment: $$X\sim \Gamma(\alpha, \beta), 1\le k < \alpha$$ $$ E(X^{-k})=\frac{\beta^k}{\prod^k_{i=1}(\alpha-i)} $$ And I wonder what would happen if $k$ is ...
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Let $U=(u_1, u_2)$ and $V=(v_1, v_2)$ be two randomly distributed points on the Euclidean plane assuming bivariate normal distributions $U \sim N(\mu_u, \Sigma_u)$ and $V \sim N(\mu_v, \Sigma_v)$ with ...
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Let $𝑍=𝑋𝑌$ , where $X$ and $Y$ are independent, 𝑋 ~𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙(0.01) and 𝑌∼𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖(0.3) Is there a way to find the m.g.f of 𝑍? I know that I can find the C.D.F by doing as ...
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So the initial question was: Let X and Y be independent random variables with common moment generating function $m_X(s) = m_Y (s) = e^{s^2/2}$. a) Determine the moment generating function $M_V (s)$ of ...
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I'm supposed to show that all of the moments of the density $\text{exp}(-|x|^{1/2})$ are finite. I'm not convinced this is true though. The $p$th moment is \begin{align*} \mathbb{E}[X^p] &= \int_{-...
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A sum of logbeta distributed variables occurs in this question Distribution with a given moment generating function Let, $X_j \sim Beta(j\sigma, 1-\sigma)$, $Y_j = -\log(X_j)$ and $S_n = \sum_{j=1}^n ...
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Let $X_1,X_2,\dots,X_n\stackrel{iid}{\sim}Poiss(\lambda)$ Let mgf of $X_1$ is given by $M_X(t)=e^{\lambda(e^t-1)}$ and let $\bar{X_n}=\frac{1}{n}(X_1+X_2+\dots+X_n)$ Then, by Weak Law of Large Numbers ...
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I am trying to derive the moment generating function for the absolute value of a Skellam random variable $Skellam(\lambda_1, \lambda_2)$ Suppose $X_1 \sim Pois(\lambda_1)$ and $X_2 \sim Pois(\lambda_2)...
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Let $X_{1}$ and $X_{2}$ be independent random variables with respective moment generating functions as $$M_{X_{1}}(t) = (\frac{3}{4} + \frac{1}{4}e^t)^3 \ , \ M_{X_{2}}(t) = e^{2(e^{t}-1)}$$ , $$...
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Consider a random walk with $S_n=\sum^n_{i=1}X_i$, where the random i.i.d. steps $X_i$ take values $-1,0,2$ with probabilities $1/9,1/9,7/9$ respectively. Set $S_0=1$. I would like to calculate the ...
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Studying for a test in course about stochastic processes, here's a test question that I can't fully understand: An insurance company insures its policyholders against damages of a particular kind. ...
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I am working on calculating the moment generating function for the pdf $f_X(x) = \frac{\sin(x)}{2}$ with the bounds $[0, \pi]$, and here is my attempt although I would like to know whether I have ...
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The moment generating function of a normal distribution is defined as $M(t) = \int_{-\infty}^\infty e^{tx}\frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2} dx$ In a book I’m ...
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