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

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recurrence simple random walk in one dimension before hitting time

Consider Simple Random Walk in one dimensions, starting from $x \in \mathbb{Z}^+$. The walker jumps to the right with probability $p$ and to the left with probability $1-p$. Assume $p \leq ...
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3answers
27 views

Is $E[Bin(X,p)]=E[X]p$?

We have given some random variable $X$ with mean $E[X]=:\mu$. Now we are interested in a random variable $Y \sim Bin(X,p)$. Is it true that $$E[Y]=\mu p?$$ What confuses me is that normally the ...
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60 views

Why “One cannot construct more than countably many independent random variables”?

I'm reading the book "Large Networks and Graph Limits" by László Lovász. On the page 18 he said the following: One cannot construct more than countably many independent random variables (in a ...
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1answer
57 views

Example of non continuous random variable with continuous CDF

Can someone provide an example of $X$ being a non-continuous random variable with continuous cumulative distribution function? For instance: $X$ is discrete if it takes (at most) a countable number ...
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1answer
24 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 ...
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1answer
45 views

Expectation related to Normal distribution and its density

Given $\sigma^2>0$. Let $Z\sim N(0,1)$ and $\Phi$ be the cumulative standard normal with density function $\phi$. I wish to show that $$ E\left(\frac{Z^2}{[\phi(\sigma Z)]^2}\Phi(\sigma ...
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2answers
54 views

let $X$ be a standard Gaussian random variable. Show that $(X,X)$ is not absolutely continuous.

i'm trying to understand a proof of the following statement: let $X$ be a standard Gaussian random variable. Show that $(X,X)$ is not absolutely continuous. I'll write down the proof in such a ...
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1answer
19 views

Sum of binomial distributed random variables

Let $X \sim Bi(n,p), Y \sim Bi(m,p)$. “Visual arguments” suggest that $X+Y \sim Bi(m+n,p)$. However, I am unable to prove that. Using the definition I can reduce the problem to $$\sum_{i=0}^k ...
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0answers
28 views

Problem calculating the average power of a vector?

I am calculating the average power of a vector. I would like to compare the final expression with the simulation. However, they are not equal. Please help me to point out which steps are wrong. Thank ...
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1answer
40 views

probability of getting 5 calls in 5 minutes

Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways. 1 a. Compute the probability of receiving three calls in a 5-minute interval of time. b. Compute the ...
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1answer
31 views

Box-Muller method for correlated normals

The standard Box-Muller method produces two independent normal variables given two uniform ones. Is it possible to extend the method such that given a correlation coefficient $\rho\in[-1, 1]$ and two ...
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2answers
32 views

On the definition of a random variables

Let $(O,F,P)$ be a probability space. That is $O$ is a set, $F$ is a $\sigma$-algebra of subsets of $O$ and $P$ is a probability measure. Consider a function $f:O\to\mathbb R$. Would we call $f$ a ...
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1answer
60 views

Operations on Random Variables

It is known that the equivalent resistance of a parallel combination of two resistors is equal to \begin{align*} R = \frac{R_1R_2}{R_1+R_2} \end{align*} which could be also written as ...
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0answers
20 views

For a sequence of random variables with bounded probability density function, can their joint pdf be unbounded?

For $d\geq2$, let $X_{i}=\left\{Y_{i-1},Y_{i-2},...,Y_{i-d} \right\}$, and assume the sequence $\left\{X_i \right\}$ is strictly stationary. Let $f_{j}(x_{0},x_j)$ denote the joint density of ...
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1answer
56 views

Prove Number of Arrivals $N(s)$ up to time $s$ follows $\mathrm{Poisson}(\lambda s)$ Distribution

This comes from my self-study of Durrett's "Essentials of Stochastic Processes" book, page 97. Definition Let $\tau_1,\tau_2,\ldots$ be independent $\mathrm{exponential}(\lambda)$ random variables. ...
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0answers
50 views

More on transformations and convolution on continuous random variables

This question is related to my last question but I've done some more exploring and then got stuck again. I decided to modify the problem a little bit and use a transformation of a random variable that ...
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0answers
24 views

Distributions with infinity variance.

I'm looking for a list (or something like that) of distributions with infinity variance (or infinity second moment), like non-gaussian Stable Distributions. I have an important warning: Some ...
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2answers
37 views

Finding mean from die probability

Example 4.4.5: Suppose that there is a 6-sided die that is weighted in such a way that each time the die is rolled, the probabilities of rolling any of the numbers from 1 to 5 are all equal, ...
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0answers
8 views

Difference in magnitude between two cross-correlations by two different way of calculations.

I think there are two ways of calculating cross-correlations for two difference random variables, X and Y. I am assuming discrete functions. 1) Multiplication $$ \sum_{m=-\infty}^\infty x[m]y[m+n] ...
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1answer
24 views

Are these variables correlated [closed]

Given that $a,b,c,d,p,q$ are independent random variables, are $X=ab-cd$ and $Y = cp - aq$ correlated? If not, are they independent?
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1answer
70 views

[Probability]need help to understand the following expression

So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively. assume c is a constant. I am having huge trouble to understand the integration of the following ...
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1answer
14 views

quick question on measurability of random variables and what becoming a deterministic function means.

we stated a theorem in class: if X r.v. is $\sigma(Y)$ measurable then X is a function of Y, where $\sigma(Y)$ signifies the sigma algebra of Y. This is fine. The Professor sometimes states that X ...
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0answers
18 views

Kullback-Leibler or Jensen-Shannon divergence between two distributions.

i would like to understand well what Kullback-Leibler or Jensen-Shannon divergence between two distributions will tels us about two distribution,for instance let us consider following code ...
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15 views

Are functions of independent random variables related to each other by a constant independent

I have $6$ random variables $a,b,c,d,f,g$, each having independent Gaussian distribution. Now I define following three random variables \begin{equation} X = ab - cd\\ Y = cf - ag\\ Z = gd - bf ...
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1answer
45 views

Summing dependent random variables with unknown joint cdf

Suppose that X_1, X_2,... X_5000 are discrete and dependent non-identically distributed random variables, whose marginal distributions are known, but whose joint distribution is not known. Is there ...
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17 views

exponential inequality for sum of dependent random variables

I have proved an inequality for the expectation in the context of dependent random variables. Can you please confirm it and give me some feedbacks? If $X_1,X_2,X_3,\ldots,X_m$ are $m$ dependent mean ...
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3answers
74 views

Finding expected value??

In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the ...
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30 views

Probability: NEED HELP to Understand with the follow [duplicate]

I need help to understand the probability derviation of a paper. Please help me. For the following, please only treat $|h_{R,B}|^2$ and $|h_{A,R}|^2$ as random variables (other parameters can be ...
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1answer
26 views

Product of 2 random variables:domain of integration

I am trying to compute the PDF of the product of two ind. random variables: $Z=XY$, where $0\leq x \leq d$ and $ 0\leq y \leq 1 $. ($0<d<1$) I found this formula : $ f_Z(z) = ...
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1answer
53 views

Prove that E(X) exists if and only if E(|X|) exists.

I found this theorem in a book, but there is no proof there: If X is a random variable, then Prove that E(X) exists if and only if E(|X|) exists. where $E(X)$ is the expected value of $X$ I know ...
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Almost surely convergence with stationary random vectors

I dont seem to be able to incorporate the stationarity condition into any of limit theorems I know. I cannot see how the Birkhoff almost everywhere ergodic theorem could be used as I cannot see how ...
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2answers
42 views

Show that the shift map is measurable and measure-preserving

Show that the shift map $\theta$ of Definition 6.3 is measurable and measure-preserving. Not sure how to represent $\theta^{-1}$ which I believe is where I am stopped in solving this problem.
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1answer
89 views

Generating random variates in Excel

I am very confused with a question I have found in relation to Excel. I am hoping someone can help me do this or at-least give me direction in which I can figure out how to do this. So far I don't ...
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1answer
21 views

Creating a bivariate distribution from two independent variables

If you have two random variables that are independent say $X\sim f_X (vars)$ and $Y \sim f_Y (vars)$. Is this a way to produce a bivariate distribution $f_{(X,Y)}$? $f_{(X,Y)} = p(X=x \cap Y=y) = ...
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22 views

Independent of random variables.

When reading Shiryaev's Probability. In the chapter 1, section 4. problem 11: Show that the random variables $\xi_1,\cdots,\xi_n$ are independent if and only if ...
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14 views

Could you explain intuitively the mean & variance of Chi-square random variable?

$X=X_1^2 + X_2^2 + ... +X_n^2$ , where each $X_i , i=1,...,n$ is i.i.d and and Gaussian distributed with mean $0$ and Varaiance $\sigma^2$. Why is mean of $X$ is $n\sigma^2$ and variance is ...
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1answer
21 views

Sampling on Axis-Aligned Spherical Quad

Given spherical coordinates on a unit sphere, imagine a spherical quad defined by two ranges $[\phi_0,\phi_1]$ and $[\theta_0,\theta_1]$. If you have a globe, for example, the grid formed by the ...
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2answers
23 views

Expected Value with Parameter p

The random variable X has the following probability distribution: P[X=-1]= (1-p)/2 P[X=0]= 1/2 P[X=1]= p/2 The parameter p satisfies the inequality $0 < p < 1$. Find the expected value and ...
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4answers
38 views

Generate random numbers in a random fashion

How can I generate 9 random numbers between 1 to 9,without repetition, one after another. Its like: Lets assume that the first random number generated is 4, then the next random number has to be in ...
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1answer
32 views

Normalizing constants for Extreme value distributions

I have a question regarding the normalizing constants $\mu$ and $\sigma$ that appear in the following problem. Let the random variable $Y_n$ be $Y_n=max(a_1,a_{2},\cdots, a_n)$ and $X_{n}$ be ...
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2answers
87 views

Expectation of random variables ratio

Let $X_1, X_2, \dots, X_n$ be $n$ positive iid random variables. Then show that $$E\left(\frac{\sum_{j=1}^k X_j}{\sum_{i=1}^{n} X_i}\right) = \frac{k}{n}.$$ Because of the linearlity of the ...
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1answer
88 views

If $X_{i}$ are I.I.D and $n^{-1}\sum_{i=1}^{n}X_{i}$ converges a.s/in-distribution to a constant $a$ is it true that $a=\mathbb{E}[X_{i}] $?

The question itself is in the title. It is immediate by the strong law of large numbers that if $X_{i}$ had a finite first moment then we would have a.e convergence (and thus in probability and in ...
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1answer
14 views

restricting range of values that random variable can take

I have a random variable $Y$ such that $Y=X+\epsilon$ where $X$ is not random variable and takes values in $(0,1)$. $\epsilon$ is random variable with $E(\epsilon)=0; Var(\epsilon)=\sigma^2$. What ...
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0answers
27 views

Product of stochastically independent random variables

Let $X, Y, Z$ be three stochastically independent random variables that are quadratic integrable (quadratintegriertbar is the German term, I didn't find a exact translation). No which statements hold ...
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1answer
52 views

Finding tight upper/lower bounds for $\mathbb{E}[\frac{1}{1+X^{2}}]$ where $X$ is a RV with $\mathbb{E}[X]=0$ and $\mbox{Var}(X)=\nu<\infty $

The question is pretty much in the title. My first thought was using Jensen's inquality to get some sort of lower bound. Since $\frac{1}{1+x^{2}}$ is convex on ...
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0answers
57 views

Uncountable family of random variables

Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space $(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...
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1answer
25 views

$X_i\sim \operatorname{Ber}(\theta_i)$ and $Y = \sum X_i$, sum of independent Bernoulli trials with different $\theta_i$. Find $\operatorname{Var}(Y)$

$X_i\sim \operatorname{Ber}(\theta_i)$ and $Y = \sum_{i=1}^n X_i$, sum of independent Bernoulli trials with different $\theta_i$. So this is something like we have a collection of $n$ possibly ...
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1answer
30 views

Show that $\int_Ax^2\mu_X(dx)\le\frac{12}{11u^2}\{1-\Re(\Phi_X(u))\}$

Let $A=[-\frac1u,\frac1u]$, Show that $$\displaystyle\int_Ax^2\mu_X(dx)\le\frac{12}{11u^2}\{1-\Re(\Phi_X(u))\}$$ where $\Phi_X(u)$ is the characteristic function of the r.v. $X$ Hint: ...
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1answer
34 views

Numerical CDF estimation for complicated random variable

Given a combination $U$ of several random variables $X,Y,Z...$ with known distributions, what is an efficient numerical algorithm to estimate PDF or CDF of $U$, if its CDF, PDF, characteristic ...
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1answer
39 views

Show that $\Pr(S_N\in A\mid N=n)=\Pr(S_n\in A)$

Let $X_1,.\ldots,X_n$ be i.i.d. random variables and $N$ be a positive integer-valued random variable, which is independent from the sequence. If $S_n=\displaystyle\sum\limits_{i=1}^{n} X_i$, then ...