Questions tagged [moment-generating-functions]

722 questions
Filter by
Sorted by
Tagged with
130 views

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 ...
128 views

Compute the moment generating function of $Y = X_1X_2 + X_1X_3 + X_2X_3$

Suppose $X_1, X_2,$ and $X_3$ are independent and $N(0, 1)$-distributed. Compute the moment generating function of $Y = X_1X_2 + X_1X_3 + X_2X_3$. I know that any $X_iX_j$ with $i \not =j$ is a ...
145 views

Variance of a sub-Gaussian random variable

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 ...
43 views

43 views

42 views

58 views

Moment-generating function of $Z:=X_1X_2+X_3X_4$

Let $X_1,X_2,X_3,X_4$ be four indipendent random variable with normal distribution of mean 0 and variance 1. The exercise asks me to calculate the moment-generating function of $X_1X_2$. I was able to ...
61 views

Finding moment generating function from a given probability mass function

Let $Y_1$ and $Y_2$ be two independent discrete random variables such that $p_1(y_1) = \frac13$; $y_1 = -2, -1, 0$ and $p_2(y_2) = \frac12$, $y_2=1,6$. Let K = $Y_1 + Y_2$. Find the moment ...
102 views

38 views

Moment-generating function of $m$ independent variables [closed]

Let $X_1,...,X_n$ be independent variables, each of them has a Discrete uniform distribution between $0$ and $m$, $m= \left( 2,3,4,,... \right)$. Let $Y$ be a random variable which is defined by ...
42 views

Find the distribution of $Z=X+Y$ where both $X$ and $Y$ are exponentially distributed.

I have a problem where, in order to solve it, I need to find the distribution of $Z$. Say that $X\sim\text{exp}(\lambda)$ and $Y\sim\text{exp}(\mu)$. I don't want to use the convolution formula but ...
27 views

How to find moments/compute this integral?

I have a steady state distribution which is of the form $$K[A+Bz]^{C}e^{D(1-z)}$$ where $A,\ldots,D$ are constants. I want to find moments of $z$. I do not know how I might go about it so that I can ...
46 views

Squared Brownian motion and its moments

I have the following $X_t$ which satisfies: $X_t=a \cdot t+b \cdot W_t$ where $a,b \in R$ and $W_t$ is a Wiener process such that $W_t$ is normally distributed with $N(0,t-s)$ for $t>s$. Suppose ...
74 views

Cumulants vs. moments

In high order statistics, what is the intuition for the difference between cumulants and moments? What does any of them measure and what is the intuition to use one of them over the other? ...
40 views

What are the interest of the moments of a random variable?

Let $X$ a random variable. We define the moment of order $r\in\mathbb N$ by $m_r=\mathbb E[X^r]$. I know that the moment of order $1$ is the expectation, of order 2, one can get the variance, of order ...
66 views

53 views

121 views

Given joint moment generating function (mgf), calculate $P(X + 2Y < 2X − Y)$

Given $X+2Y$ and $2X-Y$ are independent, and that $M_{X,Y}(t,u)=\exp\left[2t+3u+t^2+\dfrac{4}{3}tu+2u^2\right]$, how would one calculate $P(X + 2Y < 2X − Y)$?