1
vote
1answer
20 views

Expected length of a random vector

I meet a basic definition about the expected length of a random vector when reading a paper: What is "expected length" How to roughly derive both equations (yellow part) (Is that Gamma ...
1
vote
1answer
25 views

How can I show that a r.v. with cumulative distribution is continuous?

I want to show that, if $F_X$ is the cumulative distribution function of a random variable $X$, then $X$ is absolutely continuous iff $F_X \in C^1(\mathbb{R})$ ? I know absolutely continuous means ...
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 $Ω=[0,1]$ ...
2
votes
2answers
103 views

Monte Carlo Importance Sampling

I am reading the book on Monte Carlo by Sobol (A Primer for the Monte Carlo Method). In the section on Importance Sampling, he writes: $I = \int_a^b g(x) \: dx$ "to compute this integral, we could ...
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 ...
0
votes
1answer
66 views

Quick Question Integration with Joint PDF

Let $X_1, X_2, \ldots, X_n$ by independent and identically distributed random variables with probability density function (pdf) $$f_X(x) = \left\{\begin{array}{ll}1, & 0 < x < 1\\ 0, ...
1
vote
0answers
46 views

Second moment of random variable in the integral form

Let $X_1,\dots,X_n$ are i.i.d. samples from uniform distribution on $(0,1)$. Let $\hat F_n$ be their modified empirical distribution function defined by $$ \hat ...
1
vote
0answers
54 views

Minimum of N Chi-square random variables when N is large

I have a problem in numerically evaluating the PDF of $Y=\min(X_1,X_2,\cdots,X_N)$ where $N=\binom{M}{K}$, the binomial coefficient and $X_i$s are iid Chi-square random variables. The CDF of $Y$ is ...
2
votes
1answer
61 views

Calculating joint MGF

This is an end-of-chapter question from a Korean textbook, and unfortunately it only has solutions to the even-numbered q's, so I'm seeking for some hints or tips to work out this particular joint ...
2
votes
0answers
29 views

Evaluate spatial variation of density-like scalar

Apologies if this has been asked previously, but I'm not totally sure of the best way to pose the question. Background I'm evaluating the variation of a spatially varying scalar field $p$ ...
1
vote
1answer
134 views

Find cumulative distribution function of a continuous random variable.

$X$ is a random variable with density $f(x)=0.5e^{-|x|}, (-\infty<x<\infty)$. Find c.d.f of $x^2$. I dont quite get the hang of these. I tried for just x and got the following. for $x<0$: ...
0
votes
0answers
156 views

What is the Dirac mass on measure space?

I am reading the book "Lectures on Stochastic Analysis." But I know seldom about measure space. I meet with a symbol which the author call Dirac mass(in 9.3 of this book). Let E be a measurable space, ...
-1
votes
2answers
129 views

How do you take prove a probability function given 2 unknowns?

I am trying to prove the probability density function that the below equals one. $$ f(x) = (1 + \alpha x)/2 $$ Given: $$-1 < x < 1 $$ $$ -1<\alpha<1$$ These < are less than or equal ...
0
votes
2answers
119 views

Joint distribution of U = X + Y and V = X - Y

I have two independent continuous random variables, X and Y, which are uniformly distributed over the interval [0,1]. From this I have two further random variables, U and V, which are defined as U = X ...
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 ...
1
vote
0answers
62 views

Let $X$ be an integrable RV with median $m$. Show that $m=\text{argmin}_a\mathbb{E}|X-a|$

I have tried to look for mistakes in these calculations until I turned blue, but still clueless. If anyone could help me pointing out the errors, I would be really grateful. My apology for the long ...
0
votes
0answers
78 views

Integration by part for Lebesgue integral

I tried to prove one theorem in convergence of random variables and found myself in a little bit of trouble when doing integration by part. The reason being it involves a Lebesgue integral which I am ...
4
votes
4answers
2k views

What does it mean to integrate with respect to the distribution function?

If $f(x)$ is a density function and $F(x)$ is a distribution function of a random variable $X$ then I understand that the expectation of x is often written as: $$E(X) = \int x f(x) dx$$ where the ...
0
votes
1answer
27 views

Stuck on 'differentiating the integral from above' for computing of a PDF

I am stuck on a math derivation that has to do with statistics, so I am putting the statistical context here for context. In short, I am stuck on understanding how the answer to the PDF was attained. ...
3
votes
1answer
391 views

Determining boundaries of Probability Density Function integral for a requested probability

This isn't one specific homework question, but a concept I'm having trouble with in class. We were asked on a couple of questions recently on homework dealing with the probability density function of ...
0
votes
1answer
124 views

Find the distribution function of X.

Let the point (u, v) be chosen uniformly from the square 0<=u<=1, 0<=v<=1. Let X be the random variable that assigns to the point (u, v) the number u+v. Find the distribution function of ...
0
votes
1answer
149 views

Integration by Parts to find variable k (PDF problem)

I am trying to do a PDF problem where $f(y) = (ky^4)(1-y)^3$ when $0 \le y \le 1$, and $f(7) = 0$ elsewhere. I need to find $k$, and currently have this: $$\int_0^1 (ky^4)(1-y)^3 ~dy = 1.$$ I'm ...