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

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25 views

What is the distribution of sum of the squares of k dependent standard normal random variables?

It is known that the sum of the squares of k independent standard normal random variables is chi-squared distributed, what happens if we look at the sum of the squares of k dependent normal variables? ...
4
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1answer
67 views

Numerical evidence of law of iterated logarithm (random walk)

The law of iterated logarithm states that for a random walk $$S_n = X_1 + X_2 + ... X_n$$ with $X_i$ independent random variables such that $P(X_i = 1) = P(X_i = 1) = 1/2$, we have $$\limsup_{n ...
0
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1answer
32 views

Expectation and Variance [closed]

A day trader buys an option on a stock that will return \$100 profit if the stock goes up today and lose \$200 if it goes down. If the trader thinks there is a 75% chance that the stock will go up, ...
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2answers
33 views

Probability that 4/4, 2/4 Discrete Uniform RVs aren't equal

If I have four discrete $uniform(1,32)$ RVs. I'm trying to figure out the probability that a) None are the same b) exactly two are the same I thought that the probability that none are the same ...
1
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2answers
68 views

Geometric distribution achieves maximum entropy for given mean

Let $X$ be a random variable with geometric distribution, ie $P(X=k)=p(1-p)^k$. If I calculated it correctly, $X$ has mean $E(X)=\frac{1-p}p$ and entropy $H(X)=-\log p - \frac{1-p}p\log{(1-p)}$ ...
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0answers
21 views

Variance in the sum of batch-correlated residuals in a regression

I am looking at a regression model of the following form: $Y=intercept+\beta_{Yf.n}X_f+\beta_{Yn.f}X_n +error$ where $X_f$ and $X_n$ are predictors. A value for $Y$ will be sampled from the ...
4
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2answers
38 views

Probability of every ball occurring in multiple independent random samples

An urn contains 5 distinct numbered balls. You choose 2 without replacement. You then reset the urn and choose another 2 without replacement. Do this one more time. Now you have three random samples ...
2
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0answers
44 views

$X_{n}$ independent and there is $a_{n}\to0$ s.t $\lim\limits _{m\to\infty}a_{m}\sum_{n=1}^{m}X_{n}$ is finite w.p 1. Then the limit is constant.

I'm trying to prove the following claim: Suppose $X_{1},X_{2},...$ are independent and there exists a sequence $\left\{ a_{n}\right\} _{n\geq1}\subseteq\mathbb{R}$ s.t $\lim\limits ...
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1answer
54 views

Prove the conditional expectation for iid binary random variables/.

Suppose that $(X_1, X_2, \dots)$ are independent identically distributed binary variables that take on the values $0$ and $1$ with probability $P[X_i = 1] = p$, $0 < p < 1$. We take a new r.v., ...
0
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0answers
41 views

heuristic for expected number of visits random walk

What is the heuristic argument that explains why, on $\mathbb{Z}^d$, $d \geq 3$, the expected number of visits of a random walk starting from the origin at $x$ is of order $$ O(|x|^{2-d})? $$
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3answers
354 views

Maximum of a sum of random variables

Let $X_1, \dots, X_n$ be independent and identically distributed random variables with $E(X_i) = 0$ and $$S_k = \sum_{i \leq k} X_i$$ What is the probability distribution of $M_2 = \max \{ X_1, ...
0
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1answer
32 views

Cumulative distribution functions and random variable problem

So the question is: If X is a random variable with a cdf $FX (t)$, and Y is the random variable given by Y = aX + b. Express the cdf $FY (t)$ of Y in terms of $FX (t)$. (Consider separately the ...
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0answers
19 views

Sub-Gaussian Random Variable with Small Variance

Write $X \in sG(\sigma^2)$ if $X$ is sub-Gaussian of parameter $\sigma^2$, that is $\mathbb{E}(e^{\lambda X}) \le e^{\lambda^2 \sigma^2 / 2}$. I'm interested in showing that, given $\epsilon > 0$, ...
1
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2answers
63 views

how to calculate expectation $X$ continuous and $Y$ discrete

if has density $f(x,y) = \dfrac{12}{13}x^y,\quad 0<x<1, \quad y=1,2,3$, how to calculate this expectation $E(Y\mid X=x)$? I am confused because $X$ be continuous random and $Y$ be discrete ...
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0answers
81 views

Infinite sum of indicators almost sure convergence

Let $S_{n}:= \sum_{i=1}^{n} X_{i,n}$ where for each $n, X_{1,n}, X_{2,n},..., X_{n,n}$ are sequences of independents r.v.'s. $$X_{i,n}=\begin{cases}1, & \text{with probability }p_n\\0,& ...
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1answer
42 views

Computing the expectation of a Tricky R.V. (Brought form Neuroscience).

I need to compute $\Bbb{E}(\tau^{X} \ \Bbb{1}_{\{\tau^{X}<+\infty\}})$ where: $1) $ $\tau^y$ is a r.v. representing the time spent by a particle until it "jumps", ( $ y \in R_{\geq 0} $ is the ...
2
votes
1answer
34 views

Convergence in probability implies Fatou's lemma?

Let $(\Omega, \mathcal{F},P)$ be a probability space and $(X_n)$ be a positive-valued sequence of random variables on $\Omega$. We assume that $(X_n)$ converges in probability to the random variable ...
1
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1answer
28 views

Almost everywhere equality of r.v.'s , based on information on mean values.

Let $X, Y$ be two random variables on a probability space $(\Omega, \mathcal{F},P)$, where $\mathcal{F}=\sigma(\mathcal{E})$. We assume that: $\mathcal{E}$ is closed on intersections, i.e. $A\cap ...
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1answer
35 views

Does this sum of normally distributed random variables necessarily result in a continuous R.V?

Originally I had asked whether two continuous random variables can sum to a discrete random variable. More specifically, I am wonder whether, if we Let $X_n \sim \text{iid } N(0,\sigma_x^2)$ and $Y_n ...
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0answers
6 views

Signing *change* of probability that one random variable is lower than another

Let $\tilde{z}_L \in [0,1]$ and $\tilde{z}_H \in [0,1]$ denote two random variables, with $F_L(z|\theta) := \Pr\{\tilde{z}_L \leq z|\theta\}$ and $F_H(z|\theta) := \Pr\{\tilde{z}_H \leq z|\theta\}$. ...
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2answers
31 views

zero covariance but not independent - normally distributed random variable $X$ and $X^2$

This is one of my homework question, which the answer sheet has already been given out. However, I still don't understand it. Exercise 1.1. It is well known that for two normal random variables, zero ...
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0answers
16 views

Cracking any linear congruential generator

I have a linear congruential generator $X_{n+1} = (aX_n + b) \bmod 2^k $with given arguments and number $Y$. The problem is to find the smallest $i$ that $X_i = Y$ or tell that there is no such $i$. ...
0
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2answers
34 views

IID variables in statistics and real-life assumptions

IID (Independent and Identically Distributed) Random variables are often used in statistics, where a truly random sample is assumed to be made of IID variables. I'm studying basics of statistics (as a ...
1
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1answer
33 views

Bound on variance of bounded random variable

For a bounded random variable $X \in [a,b]$, we know $\operatorname{Var}(X) \le (b-a)^2/4$, see for example this answer. I am trying to give an alternate proof using symmetrization. If $Y$ is an ...
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2answers
27 views

Find $Z=X+Y$ given $f(x,y) = 2(x+y)$ for $0\leq x \leq y \leq 0$ using transformation method.

Given the joint pdf $$f(x,y) = \begin{cases} 2(x+y) & 0 \leq x \leq y \leq 1, \\ 0 & \text{otherwise}. \end{cases} $$ Use the cdf transformation to find pdf for $Z = X+Y$. The ...
1
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2answers
56 views

Compute $\mathbb{E}(U_{1} | M)$ and $\mathbb{P}(U_{1}=M)$ [closed]

I am having trouble getting started on and finishing this problem. Any help that can be offered would be greatly appreciated. Let $U_{1},...,U_{n}$ be independent random variables uniformly ...
1
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2answers
70 views

Compute $P(X_{1}=2\mid X_1+X_2+\cdots+X_{n})$ for $(X_i)$ i.i.d. Poisson distributed [closed]

The problem says: Let $X_{1}, X_{2}, ..., X_{n}$ be i.i.d. Poisson random variables. Let $S_{n}=X_{1}+...+X_{n}$. Compute $P(X_{1}=2\mid S_{n})$. I'm trying to solve this problem. I'm really ...
1
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0answers
13 views

fidi of Chi-square Random Field

The $\chi^2$ random field $U(t)$ with $n$ degree of freedom (dof) is defined as: \begin{align} U(t) = \sum_{i=1}^n X_i(t)^2, t\in\mathbb{R}^N \end{align} where $X_1(t),...,X_n(t)$ are i.i.d ...
0
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2answers
23 views

convergence in distribution of truncated gaussian variables

Let $X$ be a random variable which is distributed normally with mean $\mu=0$ and variance $\sigma=1$. Suppose that $X_n$ is a random variable for any positive integer $n$ with truncated normal ...
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2answers
26 views

Effect on probability of adding a constant to the random variable

I have this question in my notebook.A Drunk person performs a random walk over positions $0,\pm1,\pm2,\dots$ He starts at 0, he takes successive 1 unit steps going to the right with probability p and ...
0
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1answer
70 views

Is the expected value of the difference of these two random variables, with infinite expected value, $0$, or undefined?

Let's say we have two independent random variables, $x_1$ and $x_2$, both have a probability mass function $X$ defined as $$X(n) = \begin{cases} 2^{-m} & \text{if $n=2^m$ for $1 \le m \in \mathbb ...
3
votes
1answer
27 views

Finding a random variable $X$ such that $X_n$ (given) converges in distribution to $X$

For every $n\in\Bbb{N}$, let $X_n$ be a random variable which gets the values $\{-1, -\frac{n-1}n,...,-\frac 1 n, 0, \frac 1 n,...,\frac{n-1}n, 1\}$ with equal probability. Find a random ...
0
votes
2answers
36 views

CDF from PDF when $P(X \ \text{is} \ \text{even}\mid X\geq4)$

Given a PMF p by x 2 3 4 5 6 p(x) 0.1 0.2 0.2 0.3 0.2 And let X be a random variables with values in the set {2,3,4,5,6} Is it correct to ...
1
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0answers
31 views

Random Samples and Sample Variance Bound

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from a population. Show that: $$\max_{1 \leq i \leq n}|X_{i}-\bar{X}|<\frac{(n-1)}{\sqrt{n}}S$$ Where we have the sample variance ...
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0answers
25 views

Geometry of Vector Random Variables and Joint Distribution

I'm not a statistician but have been trying to understand the following problem in my research: I have two $3\times 1$ random vectors $\mathbf{v}$ and $\mathbf{w}$, and a function ...
0
votes
1answer
21 views

Calculate $P(X\leq 1\mid Y\leq0)$

I need to calculate $P(X\leq1\mid Y\leq0)$ I've found that $$P(X\leq1\mid Y\leq0)=\frac{P(X\leq1,Y\leq0)}{P(Y\leq0)}$$ But is it true that ...
0
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2answers
37 views

Doubt in Conditional Probability

I'm studying Information theory from the book Information Theory, Coding and Cryptography-Rajan Bose. I got confused at one pos where they have derived the equation ...
0
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0answers
23 views

Implications of symmetric probability density function

Consider a real-valued random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with probability density function $f$. Suppose $f$ is symmetric around zero. This ...
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0answers
26 views

Bernoulli Random Variables with Pairwise Negative Correlation

I was wondering if there is a simple way to generate Bernoulli Random Variables that have negative correlations pairwise with a lower bound on the success probability? If that isn't possible, then ...
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0answers
27 views

How can I compute the number of selected green ball from a given selection prob.

I have $2$ red balls in box 1 and $4$ green balls in box 2 as figure. The prob. selection the red balls (R) in box 1 is $$P(R=1)=0.1$$ $$P(R=2)=0.9$$ And prob. selection the green balls (G) in box ...
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0answers
29 views

Expected length of a path in a hyper graphs

Assume in a graph we represent each node by k bits. Rout is a path that connect a node with its destination and is unique. for example, from node 000 to node 111 we need a rout with length 3 and that ...
1
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0answers
52 views

Convergence of a sum of random variables with Bernoulli coefficient.

I present a problem which is connected with some of my previous questions. Suppose that $Y_t$ is a "regular enogh" (for example $Y_t=W_t$ with $W_t$ a Brownian motion) stochastic process with ...
1
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0answers
20 views

calculation of the maximum likelihood estimator

I am given n random variables $ y_1 = \theta_1 \theta_2 + e_1$ $ y_i = \theta_1 + e_i$ for $ \space i = 2,...,n$ where $\theta = [ \theta_1 \space \theta_2] ...
1
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1answer
30 views

Limit of Snedecor's F

Suppose we have a random variable $X$ such that $X\sim \dfrac{d}{n-d}F(d,n-d)$, with $d,n\in\mathbb{Z}$. What happens when $n\to\infty$? And when $d\to\infty$? I think when $n\to\infty$ then it goes ...
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0answers
18 views

Convergence of a sequence of Bernoulli variables.

For $\lambda\in(0,1)$ consider the following sequence of Bernoulli random variables $$ \mathbb{P}\left[B_n=1\right]=1-\frac{\lambda}{n},\quad \mathbb{P}\left[B_n=0\right]=\frac{\lambda}{n}. $$ Now ...
0
votes
0answers
36 views

Gcd and i.i.d. random variables

As I was working for my test on random variables I ran into the following problem that I can't solve : Let $X$ and $Y$ i.i.d. following the discrete uniform distribution for n a natural integer. ...
0
votes
1answer
27 views

Sum of transformations of continuous uniform random variable

Let $X$ be uniformly distributed on $(a,b)$. I want to find the cdf of $$ \sin^2(X) + \cos^2(X) $$ My feeling is that since $\sin^2(X) + \cos^2(X) = 1$, the cdf will be: $$F(1 \le x)= \begin{cases} ...
1
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1answer
17 views

Transformation related to a particular random variable

Consider a random variable $Y$ with the density function given by $$f_Y(y) = \left\{\begin{array}{cc} 2y^3 & -1 \leq y \leq 1 \\ 0&\text{otherwise.} ...
1
vote
1answer
44 views

Find the maximum and minimum value of variance. [duplicate]

Let $X$ be an arbitrary random variable takes values in $\{0,1,2,...,10\}$. Then the minimum and maximum values of variance of random variable $X$ are $0$ and $30$ $1$ and $30$ $0$ and $25$ $1$ ...
0
votes
1answer
17 views

Probability of terms involving dependent random variables

I want to calculate the probability as \begin{equation} \mathcal{A} = \mathbb{P}\left(B X > \zeta_s, CXY > \zeta_s\right), \end{equation} where B, C, and $\zeta_s$ are positive constants, while ...