Questions about maps from a probability space to a measure space which are measurable.

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2
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2answers
106 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 ...
-3
votes
1answer
58 views

Find the expectation $E[X]$ [closed]

Let $X$ be a random variable which is uniformly chosen from the set of positive odd numbers less then 100. Find the expectation $E[X]$?
1
vote
1answer
29 views

A continuous random variable map by continuous function will become continuous?

Let $X$ be a continuous random variable and let $g$ be a non-constant real-valued continuous function. Prove or disprove that $g(X)$ is a continuous random variable. Note : Here it is assumed that ...
1
vote
3answers
72 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
30 views

Showing independence of rectangular events…

Suppose I have a sequence of independent random variables $\{X_n, n \in \mathbb N\}$. How do I show formally that $P((X_1,...,X_n)\in A, (X_{n+1},...)\in B) = P((X_1,...,X_n)\in A)P((X_{n+1},...)\in ...
2
votes
1answer
34 views

Convergence almost surely and B-C lemmas

Showing the expectation is straightforward. I am not sure how to use the Borel-Cantelli lemmas to show the almost surely part.
2
votes
2answers
53 views

Computing cov of 2 binomial random variables

we drop a normal cube 20 times. X - is the number of even values Y - is the number of times the cube landed on 3. As much as I understand: $$X\sim B(20, \frac{1}{2}) \\Y \sim B(20, \frac{1}{6} )$$ ...
1
vote
0answers
19 views

Reference for higher moments of nonnegative random variable as integrals of the CDF

I know how to prove $E(X^n) = \int_0^\infty \! (1 -F_X(u^{1/n})) \, \mathrm{d} u$, for a positive and continuous random variable $X$ with CDF $F_X(x)$---note that for $n=1$ it is the standard $E(X) = ...
1
vote
1answer
38 views

Does every random variable(continous) has a probability density function?

what is the criterion for a random variable(continous) for existence of probability density function for it? Could you provide some cases of random variable(continous) where pdf ceases to exist.
0
votes
1answer
30 views

calculation of variance from cdf (no mathematical expression available)

Is it possible to calculate the variance of a continous random variable from the Cummulative distributive function plot ? We dont have the mathematical expression for cdf, all we have is just a plot ...
1
vote
3answers
50 views

Relationship between Binomial and Bernoulli?

How should I understand the difference or relationship between Binomial and Bernoulli distribution?
0
votes
0answers
58 views

What is the probability density function of the cosine of a gaussian random variable?

I want to find the probability density function of $Y=\cos(X)$, where $X\sim N(\mu, \sigma^2)$. The answer is known when $X$ is uniformly distributed $U(-\pi, \pi)$ and it is an arcsin pdf, given by, ...
2
votes
2answers
51 views

Independence of random sum variables

Let $(T_i)_{i \in \mathbb{N}}$ be a family of i.i.d. random variables where every $T_i \sim\mathrm{Exp}(\lambda)$. Now let $$Y :=\sum\limits_{j=1}^N T_j$$ such that for all $1 \leq j \leq N-1$ we have ...
2
votes
1answer
31 views

Question about $L^1$ convergence for random variables

For a random variable $X \colon \Omega \to \mathbb{R}$ and a sequence of random variables $X_n$ with $$ \lim_{n \to \infty} \mathbb{E} [|X_n -X|] = 0,$$ I have found that $$ \lim_{n\to \infty} ...
1
vote
0answers
26 views

Random sampling and i.i.d.

Can you help me to clarify the following concepts by stating whether what I have written below is right or wrong? -random sampling: units are drawn from the population with a known probability of ...
0
votes
1answer
55 views

subscript notation in conditional probability

$X$ and $Y$ are two discrete random variables with joint p.m.f $p_{XY}$ such that $p_{XY}(x_i,y_j) = P(X=x_i, Y=y_i)$. I came across a notation that refers to $p_{X}(x|y)$. How do I express it in the ...
0
votes
1answer
10 views

cumulative distribution function calculation 3

given FX(X) = x^2, compute P(1/4 < X < 1/2). sorry, I am new to here so don't really know how to type them more mathematically.
0
votes
1answer
25 views

random variables function

consider flipping two fair coins. Let $X=1$ if the first coin is heads, and $X=0$ if the first coin is tails. Let $Y=1$ if the second coin is heads, and $Y=5$ if the second coin is tails. Let $Z=XY$. ...
1
vote
1answer
29 views

On the gist of $\sigma(X_1,\ldots, X_n)$

As far as I understand the reason we have $\sigma(X_1,\ldots, X_n)$ all over the probability theory is that it tells us what questions are answerable by $X_1,\ldots, X_n$. Say, we run an experiment ...
-1
votes
1answer
31 views

How to see (in)dependence of random variables based on their joint density

This is a valid joint pdf. I just want to know if X1 and X2 are dependent or independent rvs ? Why ? Thank you for your help. Is there a way of seeing this without computing the marginal density ...
1
vote
1answer
36 views

Is $\mathcal{L}^p \subset \mathcal{L}^{p-1} $?

A random variable $X$ is called integrable if $E[X] < \infty$. We say that $X \in \mathcal{L}^1$ if $E[X] < \infty$, and in general $X \in \mathcal{L}^p$ if $E[|X|^p] < \infty$. I know that ...
0
votes
1answer
29 views

number of ones with neighbours in a random binary string

Consider a sequence of i.i.d. random variables $(\xi_i)_{1 \leq i \leq L}$ such that $\xi_1 \in \{0,1\}$ and $P(\xi=1)=p$. Introduce the function $N : \{0,1\}^{L} \rightarrow \mathbb{N}$ which counts ...
0
votes
0answers
20 views

A variation of Polya's urn

Polya's urn model is as follows: you have $a\in \mathbb{Z}_{>0}$ red balls and $b\in \mathbb{Z}_{>0}$ blue balls in a urn. Suppose you pick a red ball. Then you put back $c\in ...
1
vote
1answer
92 views

Can someone please help to understand the following probability

I was reading something on communication, then I came across the following equation: $Power_{rx}=Power_{tx}*|R|^2/(1+d^2)$ where $Power_{tx}$ and $d$ can be assume to be constant, and R is the ...
1
vote
2answers
41 views

Probability Generating Functions with Three Dice

Three identical dice are thrown. The dice are fair, that is, for all three dice the probability of turning up face $j$ is $1/6$, $1 \le j \le 6$. Let $X_1,\ X_2,\ X_3$ be the independent random ...
0
votes
0answers
32 views

Expectation and Variation of dependent RVs

This is a really nice question, and while I can think of a solution to both parts, I wonder if there's a more elegant one to the latter: A fair 6-faced dice is tossed once. In a box there are 6 ...
2
votes
2answers
38 views

probability, expectation, variance

A 10-digit long number is picked randomly and each digit's pick is independent and has an equal probability of being picked (1/9 because there's digits 1 to 9). Let $X = \#\{\text{missing digits}\}$ ...
1
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0answers
24 views

Does monotone convergence theorem gives uniform convergence?

Monotone convergence theorem If $X_n$ are positive random variables and increasing to $X$, then $$\lim_{n \to \infty} E[X_n] = E[X]$$ My problem, though, is that $X$ depends on $m$, so it ...
1
vote
1answer
47 views

How does a pdf of the difference of two random variables relate to the pdf of each random variable

Let $T_1$ and $T_2$ be non-negative continuous random variables (rv) denoted in the form $T_i = \mu_i + \sigma_i X_i$ for $i=1,2$ where \begin{eqnarray*} T_{1} &=&\mu _{1}+\sigma _{1}X_{1} \\ ...
0
votes
1answer
72 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, ...
4
votes
2answers
51 views

Conditional Expectation of X given X^2

What can we say about $E[X|X^2]$ in general? And if $X$ has density $f$ respect the Lebesgue measure?
0
votes
1answer
30 views

$\sigma(Y)$-measurable R.V. $X$ and Borel functions

I have to prove that if $Y: \Omega \rightarrow \mathbb{R}$ then $X: \Omega \rightarrow \mathbb{R}$ is a $\sigma(Y)$-measurable function if and only if exists a Borel function $f: \mathbb{R} ...
1
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0answers
47 views

Moment-generating function of a generalised normal random variable

Let $X$ be a random variable that follows the "version 1" generalised normal distribution described here, with p.d.f. ...
0
votes
2answers
29 views

Probability of special configuration of ones in a binary string

Consider the sequence $(X_i)_{1 \leq i \leq L}$ of i.i.d. random variables, where $X_1 \in \{0,1\}$ and $P(X_1 =1) = p$. For a $k \in \mathbb{N}$ define the event $A_{k,L}$ as "all ones in the ...
0
votes
0answers
15 views

Two i.i.d Rvs (Gaussian)

Q: You have two i.i.d Rv's X~N(0,1) Y~(0,1). Let Z=(X+Y)^2. a) Find the mean on Z i.e E[Z}. b) Find Corr(X,Z) & Corr(Y,Z). c) Determine if Z & Y are uncorrelated. Ans: Finding E[Z] was ...
1
vote
3answers
56 views

Random Variable Problems?

Can someone show me how to work this out? I can't get the answers in the boxes.
2
votes
2answers
67 views

Almost sure convergence of a sequence of random variables

Once again I've encountered a problem, which might not be difficult: I'm given a sequence of random variables $ (X_n) $, each with density function $g_n(x) = nx^{n-1} \textbf{1}_{(0,1]} $. I am to ...
3
votes
1answer
176 views

PDF of sum of two random variables

Assume an $n$ dimensional random variable $U$ that is uniformly distributed in the volume of an $n$-sphere with radius $R$. Assume another $n$ dimensional random variable $N$ that is distributed ...
1
vote
1answer
49 views

Convergence in probability of iid normal random variables

Let $X_1, X_2,\ldots$ be a sequence of iid normal random variables with zero mean and unit variance. I read the following as a trivial example: (1) $X_n \to X_1$ in law, (2) $X_n \not\to X_1$ in ...
2
votes
0answers
27 views

Rosanov - Probability Theory Chapter 4 Question 5

I am trying to solve one of the questions in Rosanov - Probability (Chapter 4 Question 5), but I am not exactly sure what the question is asking of me. The question is: Random variable $E$ with ...
0
votes
0answers
18 views

The distribution of minmax and maxmin deviations of a Random variable

Let $X_1,X_2,X_3,......,X_n$ be $n$ independently and uniformly distributed random variables in the interval $[a,b]$. Further let $P=\min \{X_i,i=1,2,3..,n\}$ and $Q=\max\{X_i,i=1,2,3..,n\}$. ...
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votes
1answer
19 views

Finding number of points in a bounded set when number of points in the unbounded set are known.

Consider a random distribution of points in a Random 2D plane. I would like to find the number of points in a circle within this plane. Can anybody helps in solving the problem? Regards
1
vote
1answer
28 views

Borell Cantelli Application

If i got that $\mathbb{P}(\underbrace{|X_{n}|>n^{\frac{1}{2}+\epsilon}}_{=:A_{n}})\leq \exp\left(-\frac{n^{2\epsilon}}{8}\right)$ with $\epsilon \in (0, 0.5)$. I know that ...
1
vote
1answer
38 views

Random variables $x_i$ with $\lim_{k\to\infty}\frac1k\sum_{i=1}^kx_i=0$

I am looking for a sequence $(x_n)_{n\in\mathbb N}$ of random variables such that the sequence hasn't any expected value and $\lim_{k\to\infty}\frac1k\sum_{i=1}^kx_i=0$. I thought about using a ...
3
votes
2answers
180 views

Variance of a function of independent random variables

Suppose I have two discrete independant random variables $X$ and $Y$, and that I'm interested in the expected value of the random variable $W$, where: $$ W= \text{sign}(X-Y). $$ So, W is 1 if ...
0
votes
0answers
24 views

Weighted random walk in 1-dimension

Suppose we have random walker on a line, he can only stay on sites which are, say, a distance $a$ from each other. At each step he can go left or right. Every time he steps on a site, makes the ...
2
votes
0answers
35 views

$\sup_nX_n<\infty$ almost surely iff $\sum_nP(X_n>A)<\infty$

Let $(X_n)_{n\in\mathbb N}$ be a sequence of independent random variables. Show that $\sup_nX_n<\infty$ almost surely iff there exists $A>0$ such that, $\sum_nP(X_n>A)<\infty$ By ...
1
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0answers
35 views

How do I prove the special case of the central limit theorem?

Let $(X_n)$ be an i.i.d. sequence such that $\mathbb P(X_1=1)=\frac{1}{2}+\varepsilon$ and $\mathbb P(X_1=-1)=\frac{1}{2}-\varepsilon$ for some $\varepsilon\in(0, \frac{1}{2})$. I'd like to show that ...
1
vote
1answer
37 views

Prove $\{Z=0\}\subset\limsup\limits_{n}\{X_n<\epsilon\}$

Let $(X_n)_{n\in\mathbb N}$ be independent real random variables, with values in $(0,\infty)$. Consider the random variable $Z(w):=\inf\limits_{n\in\mathbb N}X_n(w)$. Prove that for every fixed ...
2
votes
1answer
38 views

Finding the distribution of $M$

Let $(X_1,X_2)$ be uniformly distributed in $[0,1]^2$ and define $Y_1=\max(X_1,X_2)$, $Y_2=\min(X_1,X_2)$. What is then the distribution of $M:=Y_1-Y_2$ ? To find the joint distribution we take a ...