0
votes
1answer
18 views

Evaluating the integral to find the expected value of the exponential random variable

I want to find the expected value of the exponential random variable. I have $E(X)=\int_{0}^{\infty }xae^{-ax}dx$. I use Integration By Parts (IBP). Let $v=-e^{-ax}\Rightarrow dv=ae^{-ax}dx$, and ...
1
vote
1answer
45 views

How to compute the integrals in inverse formula?

I have following characteristic function for certain random variable X: $$\Phi (t) = \frac{\beta_1\beta_2}{\eta_1}\frac{\eta_1 - it}{(\beta_1 - it)(\beta_2 - it)}$$ where $\eta_1 > 0, \quad\beta_1 ...
1
vote
1answer
33 views

Proving and visualizing $\mathbf 1_{(x,x+a]}(y) = \mathbf 1_{[y-a,y)}(x)$

Here is a trick from one of the proofs in probability: $$\iint \mathbf 1_{(x,x+a]}(y) \ \lambda(dx) \ \mathbb P(dy) = \iint \mathbf 1_{[y-a,y)}(x) \ \lambda(dx) \ \mathbb P(dy)$$ for $a>0$. So ...
0
votes
0answers
15 views

Probability density function of an element

How to find the probability density function of $x_m\left(1\le m\le n\right)$ from joint density function, $p_X\left(x_1,x_2,\cdots,x_n\right)$, of $n$ random variables which satisfy following ...
0
votes
0answers
31 views

A particular form of expectation

I am stucked in showing an equality concerning the expectation of a fonction $f \in \mathcal{L}_1(P) , ||f||_1 = \int_R |f(x)|dP(x)$ , where $P$ is a probability measure. The exercise is the ...
2
votes
3answers
273 views

Integral change that I don't understand

If $f(t)$ is a Probability density function of a positive RV. $\int_0^\infty\int_x^{\infty}f(t)dtdx$ Using fubini theorem should become $\int_0^\infty\int_0^{t}f(t)dxdt$ But why? Surely the answer ...
0
votes
1answer
26 views

Evaluating an expectation of the supremum of collection of random variables

I know that $\mathbb{E}(sup_n|X_n|)=\infty$ if $X_n=\frac{2^n}n\cdot\mathbf 1_{(1/2^{n+1},1/2^n)}$. However I am not sure how this can be evaluated explicitly. The probability space is $Ω=[0,1]$ ...
0
votes
1answer
73 views

Why is $fg$ integrable w.r.t. a probability measure if $f,g$ are Lebesgue integrable?

In one of the proofs, my text mentions that if $f,g$ are Lebesgue integrable then $fg$ is integrable with respect to a probability measure. I guess I have missed something, since it doesn't look ...
1
vote
3answers
70 views

Expected value of the function of a random variable

I am studying Probability and Monte Carlo methods, and it feels that the more I study the less I truly understand the theory. I guess I just confuse myself now. So the expected value of a random ...
1
vote
1answer
21 views

Integrability and exponential integrability

I'm working on a paper, and I don't know if there is some kind of typo or if I just don't get what seems obvious to the author. Note : I'll be working with probabilities, but I guess this would be ...
0
votes
0answers
20 views

Stationary distribution of Waiting Time in a $GI/GI/1$ queue

I am trying to find if there is any literature where I can find formulas for the stationary distribution of a $GI/GI/1$ queue. Specifically, I need to find $P(W=0)$ where $W$ is the steady state ...
3
votes
2answers
42 views

Substitution in integral

I am working on a probability theory excercise and encountered the following integral: $$ \iint_{(x,y)\in A}\frac{1}{2}(x+y)e^{-(x+y)}dA, $$ where $A = \{(x,y)\in\mathbb{R}^2\,:\,x+y\le z\,;\, ...
0
votes
2answers
58 views

Difficulty finding Expectation of a special function

I have a special function given as: $${\rm f}\left(r\right) ={1 \over \beta\lambda}\,2^{r/\beta} \exp\left({\left[2^{r/\beta} - 1\right]K \over \lambda}\right)$$ I should find the Expectation of ...
1
vote
0answers
23 views

Hellinger Integral properties

Let $\mu , \nu$ be two probability measures on $(\Omega , \mathcal{F}).$ Suppose we have a probability measure $\lambda$ such that both $\mu , \nu$ are absolutely continuous with respect to ...
2
votes
1answer
38 views

Absolutely Continuous measures and Hellinger integral

Let $\mu , \nu$ be two probability measures on $(\Omega , \mathcal{F}).$ Suppose we have a probability measure $\lambda$ such that both $\mu , \nu$ are absolutely continuous with respect to ...
0
votes
0answers
43 views

Change of variables in Lebesgue-Stieltjes integral

Suppose $F$ is a probability distribution function with corresponding measure $\mu_F$ on $\mathbb{R}$. Suppose moreover that $f:\mathbb{R} \rightarrow\mathbb{R}$ is a continuous, increasing function. ...
0
votes
1answer
33 views

Scaling the Lebesgue-Stieltjes integral

Suppose that $F$ is a distribution function. Denote by $\mu_F$ the measure on $\mathbb{R}$ induced by $F$. Suppose that $a>0$. Define a new distribution function $F_a$ by $F_a(x):= F(ax)$, and ...
0
votes
0answers
35 views

Proof convolution formula two stochastic variables

Let's say I have two continuous independent stochastic variables, defined on $(0, \infty)$. With densities: $X_1$ ~ $f_1(t_1), t_1 \in (0, \infty)$ $X_2$ ~ $f_2(t_2), t_2 \in (0, \infty)$ The ...
0
votes
1answer
41 views

range of a function defined by expectation

Given $c>0$ and define a function $f:(0,\infty)\mapsto \mathbb{R}$ by \begin{equation} f(\sigma)=\frac1{\sqrt {2\pi}\sigma}\int_{-\infty}^\infty\max\{-c,\min\{x,c\}\}^2e^{-\frac{x^2}{2\sigma^2}}. ...
1
vote
1answer
44 views

Limit of function defined by expectation

Given $c,\sigma,\tau$ are positive real constants and define a function $f:(-1,1)\mapsto \mathbb{R}$ by \begin{equation} ...
0
votes
0answers
10 views

Existence of invers of a function from covariance matrices space

Let $(\mathbb{R}^k,{\cal A})$ be a measurable space. Fix $c>0$ and for every $X\in \mathbb{R}^k$ define $X_c$ as $k-$dimensional vector such that the $i-th$ element of $X^{(c)}$ is $(-c)\vee ...
1
vote
1answer
111 views

Find the Method Moment Estimator of parameter $\theta$

Find the MME of parameter $\theta$ in the distribution with the density $f(x,\theta)=(\theta +1)x^{-(\theta+2)}$, for $x>1$ and $\theta >0$. So far I think I have a basic understanding of the ...
2
votes
1answer
48 views

What is the value of the following integral? ( including the inverse of the CDF of the standard normal distribution)

What is the value of the following integral? $$f(\alpha)=\int_0^\alpha [b-c \Phi^{-1}(1-\beta)] d\beta$$ where $0<= \alpha=<1$ ,b and c are constants, and $\Phi$ is the cumulative ...
1
vote
2answers
42 views

Calculate the value of c for which f is a probability density.

Let f the function defined by: Where c is positive none zero and constant . How can i calculate the value of c for which f is a probability density.Thnxs for the help.
1
vote
1answer
38 views

Empirical distribution. Problem with changing variables

We have iid random variables $X_1, X_2, \ldots, X_n$ with continuous cdf $F$. Define empirical distribution function $\hat{F}_n (x)= \frac{1}{n} \sum_{k=1}^n \mathbb{I}_{\{X_i \le x \}}$. Let's ...
1
vote
1answer
49 views

Joint To Marginal Density : Can't figure it out.

Here goes the problem: Problem: Suppose $X$ and $Y$ have the joint density function: $f(X,Y) = c \sqrt{1 - x^2 - y^2}, \,\,\,\,\, x^2 + y^2 \leq 1$ Find $c$. ...
1
vote
1answer
33 views

$X_{n}$ and $Z$ be random variables if $X_{n} \ge Z$ then $ E[\liminf_{n\to \infty} X_{n}] \le \liminf_{n\to \infty} E[X_{n}] $

Let $X_{n}$ and $Z$ be random variables on probability space $(\Omega ,\mathcal F,P)$ and $Z$ be integrable. Show that $$X_{n} \ge Z \qquad \Longrightarrow \qquad E[\liminf_{n\to \infty} X_{n}] \le ...
2
votes
2answers
59 views

I want to show if $\int |f|d\mu < \infty$ then almost everywhere $|f|< \infty$.

show that if $\int |f|d\mu < \infty$ then almost everywhere $|f|< \infty$. thanks for help.
1
vote
1answer
50 views

How do you find the distribution of this sum?

If $X\sim \text{Normal}(\mu=0, \sigma^2), Y\sim \text{Unif}(0,\pi)$, and $X \perp Y$, how do you find the distribution of $Z=X+a\cdot cos(Y)$ for some $a > 0$ ? I've found the distribution ...
2
votes
1answer
54 views

Supremum of Expectation of a Sequence of Random Variables

Let $(X_1, X_2,...)$ be a sequence of random variables that converges almost surely to a random variable $X$. Show that if $\sup_n EX_n^2 < \infty $, then $EX^2 < \infty$. I believe this is ...
0
votes
1answer
33 views

Can we make piecewise PDFs or CDFs into a single CDF?

Let $X, Y$ be independent random variables, both uniformly distributed over the interval $(0,1)$. That is, $$f_{X}(a)=f_{Y}(a) = \begin{cases} 1 & \text{if $0 < a < 1$} \\ 0 & ...
0
votes
1answer
90 views

Finding integration bounds for density of sum of two independent random variables

Let $X, Y$ be independent random variables, both uniformly distributed over the interval $(0,1)$. That is, $$f_{X}(a)=f_{Y}(a) = \begin{cases} 1 & \text{if $0 < a < 1$} \\ 0 & ...
1
vote
2answers
43 views

Generally true that $\frac{\mathrm d}{\mathrm da} \int_{-\infty}^{a-y} f(x)\, \mathrm dx = f(a-y)$?

Is it generally true that $$\frac{\mathrm d}{\mathrm da} \int_{-\infty}^{a-y} f(x)\, \mathrm dx = f(a-y),$$ where $a$ and $y$ in the expression are constants? To give context to the question, I am ...
1
vote
1answer
52 views

The intergral $I=\int _0^{\beta }f(x)dx$ is given,for $\alpha,\beta \in \mathbb{Z}$ ,how can we find $\int_0^{\alpha\beta}f(x)dx$ in terms of $I$

i am working with a gaussian normal distribution function in probability,i am given values for the integral when $z\le 4$ and i want to find a value $z=8$,in general if $z=4$ is given , how to find ...
0
votes
1answer
46 views

Finding Calculus or Probability error in basic continuous probability problem

I am having a lot of difficulty spotting my error in the following probability problem. The joint probability density function of $X$ and $Y$ is given by $f(x,y) = c\left ...
0
votes
0answers
31 views

Convolution of complex-valued probability distributions

This may be an elementary question, but I am wondering: suppose that I have two complex-valued random variables $X$ and $Y$ with corresponding density functions $f_X(x)$ and $f_Y(y)$. Obviously ...
2
votes
1answer
69 views

How to model a stochastic process, continuous in stepsize, which converges against a simple random walk?

I want to compute the probability distribution for a stochastic process with discrete number of steps, where each real value has a nonvanishing probability to be the next stepsize. And I want to ...
1
vote
1answer
48 views

Evaluation of Standard Normal Integral

I have always wondered how we calculate the percentiles of the Standard Normal Distribution given that the CDF cannot be obtained in closed form: $$F(x)=\int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}} ...
2
votes
1answer
55 views

Approximation of Stochastic integral with Stieltjes integrals

Let $V^n(t,\omega)$ be a sequence of continuous, adapted and bounded variation processes such that with probability 1, $V^n$ converges to $B$ uniformly on compact intervals of $[0,\infty)$ ($B$ is ...
0
votes
2answers
52 views

About probability density function

I have a question about probability density function in my book. It reads: A Probability density function is of the form $p(x) = Ke^{-a|x|}$ , $x \in (-\infty,\infty)$.The value of $K$ is: 1] ...
2
votes
0answers
130 views

Checking proof that a given process is a martingale

I am interested in justify the well known result about the process $M^\lambda _t =\exp\left(\lambda B_t - \frac{\lambda^2}{2} t\right)$ being $\mathcal F_t$-martingale in the filtered probability ...
0
votes
1answer
34 views

the Expectation is equal the probability of the first exit time of openset

let $X_{t}$ itô diffusion and $1_{A}(x):=\begin{cases} 1 & \text{if } x \in A, \\ 0 & otherwise. \end{cases}$ and $\tau$ be The first exit time defined by : $$ \tau=\inf\{t>0:x_{t} ...
2
votes
1answer
93 views

Weak form of Berry-Esséen theorem

Let $X$ (a real random variable) have mean zero, unit variance and finite third moment. Let $Z_{n}:=(X_{1}+...+X_{n})/\sqrt{n}$, where $X_{1}, ... X_{n}$ are iid copies of $X$. According to the ...
0
votes
0answers
65 views

Obtaining an input-output cyclic-autocorrelation relation (random process problem)

I was trying to solve an exersice in the book, "Introduction to random processes, Gardner 2nd". But I can't obtain the answer given by the book. Here is the problem: [Chapter 12 - exercise 6] ...
1
vote
0answers
189 views

On applying the Layer Cake Representation

In the answer to a previous question of mine, an anonymous user invoked the Layer Cake Representation in order to reduce proving \begin{equation} \Re\left[ \int_{-\infty}^\infty \frac{1+iy}{1+i(y-t)} ...
1
vote
1answer
74 views

show convergence of sum towards integral — not Riemann partition but using another convergence

$X_1, X_2, ...$ real random variables. $P(X_n=\frac{k}{n}) = \frac{1}{n}$ for $k\leq n \in \mathbb{N}$. Let further $X \overset{d}{=}U_{[0,1]}$. I showed $X_n \overset{d} \to X$ when $n \to \infty$ ...
2
votes
4answers
82 views

Is this Expectation finite?

How do I prove that $\int_{0}^{+\infty}\text{exp}(-x)\cdot\text{log}(1+\frac{1}{x})dx$ is finite? (if it is) I tried through simulation and it seems finite for large intervals. But I don't know ...
1
vote
0answers
54 views

Continuous random sampling with replacement.

Construct a set $s\subseteq[0,1]$ by sampling points in $[0,1]$ with uniform probability density $x\leq1$ so that $|s|=x$. Interpret this as a sampling frame during which data is captured. Now, ...
2
votes
1answer
314 views

Integral of bivariate normal distribution function with respect to itself

Define $F: \mathbb{R}^2 \rightarrow \mathbb{R}$ by \begin{align*} F(x,y)=\int_{-\infty}^{x} \int_{-\infty}^y \frac{1}{2\pi \sqrt{1-\rho^2}} exp\left(\frac{-u^2-v^2+2\rho uv}{2(1-\rho^2)}\right) ...
1
vote
1answer
37 views

Multiple random varialbles

$$f_X(x_1,x_2,x_3) = \begin{cases} \frac{1}{x_1x_2} & 0\leq x_3 \leq x_2 \leq x_1 \leq 1;\\[8pt] 0 & \text{otherwise} \end{cases} $$ question: Are $x_1,x_2,x_3$ are ...