1
vote
1answer
37 views

If $\mathbb{E}(X^\alpha)<\infty$ for $0<\alpha<1$ show that $\mathbb{E}(\min (X,t))$ is $o(t^{1-\alpha})$

If $X\geq 0$, and $\mathbb{E}(X^\alpha)<\infty$ for $0<\alpha<1$ show that $\mathbb{E}(\min (X,t))$ is $o(t^{1-\alpha})$. We know that that $$\mathbb{E}(\min(X,t))=\int_{X\leq ...
0
votes
1answer
31 views

Product of two exponentially distributed random variables

I am trying to find the close form expression of probability distribution of $Z$ such as $Z=X_1X_2$ where $X_1$ and $X_2$ are two independent exponentially distributed variables with PDF ...
0
votes
1answer
34 views

expectation over ordered random variables [duplicate]

I have a joint distribution function over random jointly distributed random variables $(X,Y)$ denoted by $f_{X,Y}(x,y)$. Assuming without loss of generality that $$X<Y$$ I would like to find ...
-1
votes
0answers
12 views

Integration over ordered random variables

I have a joint distribution function over random jointly distributed random variables $(X,Y)$ denoted by $f_{X,Y}(x,y)$. Assuming without loss of generality that $$X<Y$$ I would like to find ...
0
votes
0answers
20 views

Hypothesis testing CDF

I have the following setup. There is a set $S = \{S_1, \ldots, S_N\}$ of $N$ sensors that are probed for readings (once). Each reading is an independent sample from one of the two distributions $r_i ...
1
vote
0answers
41 views

derivative of normalizer in exponential form — change integral and gradient

When deriving the relation between normalizer and expectation of the sufficient statistic for distributions in exponential form one uses the fact, that the density integrates to one: $$1 = ...
1
vote
1answer
39 views

Prove that if expectations agree on a Pi-System, then they agree on the Sigma-Algebra generated by the Pi-System.

If $\mathscr{G}$ is a sub-$\sigma$-algebra of $\mathscr{F}$ and if X $\in L^1 (\Omega, \mathscr{F}, P)$ and if Y $\in L^1 (\Omega, \mathscr{G}, P)$ and $E(X;G) = E(Y;G)^{*} \forall G \in \mathscr{I}$ ...
0
votes
1answer
24 views

Derivative of integral over part of Gaussian distribution

I am currently trying to compute the following derivative and integral: $$ P\psi_\theta = \frac{d}{d\theta}\int_{-k}^k tf_T(t)dt, $$ where $t=x-\theta$ and $X\sim N(\theta_0,\sigma^2)$. $f_T$ above ...
0
votes
1answer
39 views

Calculate the expected value

To get the expected value of $E(X), E(Y) $ and $E(X, Y)$ given: $$ f_{X,Y}(x,y) = 3x $$ where $0\le x \le y \le 1.$ My solution is, first get the margin distribution: \begin{aligned} f_x(x) &= ...
1
vote
1answer
43 views

Proving that $ \frac{1}{n}\int_{-\infty}^{\sqrt{n}w}k(v/\sqrt{n})\phi(v)dv$ is $O(n^{-1})$

Suppose that $h:\mathbb{R}\to\mathbb{R}$ is infinitely differentiable. Define \begin{equation} k(w)=\left\{ \begin{array}{ll} \frac{d}{dw}\left(\frac{h(w)-h(0)}{w}\right)&w\neq 0,\\ ...
0
votes
2answers
53 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
16 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
33 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
277 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
28 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
92 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
89 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
25 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
24 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
44 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
24 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
40 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
52 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
34 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
37 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
46 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
121 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
49 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
50 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
51 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
66 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
35 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
102 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
45 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
56 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
33 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
73 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
50 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
56 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
55 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] ...