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

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1answer
42 views

Birthday Problem (Poisson Distribution)

I've been reading up on Poisson Distributions and have come across the following problem. My doubts are in Bold: What's the probability that in a room of n people, nobody shares the same birthday? ...
3
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1answer
22 views

statistics basic question on covariance

anyone would help me in a basic example? a fair coin is tossed, n times. X is the number of Head and Y is the number of Tails. what is the COV(X,Y).
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3answers
35 views

$X$ and $Y$ are independent and follow $U(0,1)$. Show $P(f(X) > Y) = \int_0^1 f(x) dx$

Let $X$ and $Y$ be two independent uniformly distributed r.v. on $[0,1]$, and $f$ is a continuous function from $[0,1]$ to $[0,1]$. Show that $P(f(X) > Y) = \int_0^1 f(x) dx$. I tried to prove ...
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0answers
22 views

Understanding Binomial random variables

I'm looking over Binomial random variables and I understand that $ \sum\limits_{k=0}^n k\binom{n}{k} p^k (1-p)^{n-k} = np $ from $\mathrm{Bin}(n,p)$ However, I don't understand how, if $S_n = ...
2
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1answer
24 views

Understanding Negative Binomial Random Variables

I'm trying to understand Negative Binomial Random Variables and have across the following: $ Z\sim \mathrm{NegBin}(n,p)$ if $Z = X_1 +\cdots+ X_r $ where $X_i's$ are independent identically ...
1
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1answer
11 views

Geometric Random Variable

I'm reading about Geometric Random variables from a book, which is as follows: $X_1, X_2,\ldots$ are independent identically distributed variables which are $\mathrm{Ber}(p)$ $$ Y = \min \{n\geq ...
1
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1answer
23 views

A LLN type theorem on the supremum of functions of a RV

Let $X_1,\dots,X_n$ be iid real valued random variables. Let $\mathcal{F}$ be a set of functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $\mathbb{E}f(X_i) < \infty$ for all $f \in ...
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1answer
25 views

Understanding Geometric Random Variables

I'm looking up Geometric random variables, where $X_1, X_2....$ are independent identically distributed variables which are $Ber(p)$. The book says, $$ Y = \min \{n\geq 1| X_n = 1\} \sim Geo(p)$$ ...
2
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0answers
47 views

Probability problem related to discrete random variables, binomial distribution.

I've just solved an exercise related to discrete random variables and maybe to the binomial distribution as well. I would like to know if my solution is correct, so here goes the problem statement ...
2
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1answer
60 views

If $Y\ge 0$ almost surely and $X+Y \sim X$ then $Y=0$ almost surely

Let $X, Y$ be random variables on the same probability space such that $Y \ge 0$ almost surely and $X+Y$ and $X$ have the same distribution. Please resolve whether these conditions imply that ...
1
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1answer
32 views

Conditional expectation of symmetric Sigma algebra

Another exercise with conditional expectation that I have problems with. Let $\Omega=[-1,1]$, $\mathcal{F}=\mathcal{B}(\Omega)$, $\mathbb{P}=\frac{1}{2}\lambda$. Let X be a ...
5
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2answers
38 views

Finding all Borel measures $\mu_X$ such that $Y\sim \mathcal{N}(0,1) \Rightarrow XY \sim \mathcal{N}(0,1)$.

Find all Borel measures $\mu$ on $\mathbb{R}$ such that for every independent random variables such that $X \sim \mu$ and $Y\sim \mathcal{N}(0,1)$ we have $XY \sim \mathcal{N}(0,1)$. To be honest ...
2
votes
2answers
81 views

Expected distance between two vectors that belong to two different Gaussian distributions

Let $X$, $Y$ be two random variables that follow the Gaussian distributions with mean vectors $\mu_x$, $\mu_y$, and covariance matrices $\Sigma_x$, $\Sigma_y$, respectively. The probability density ...
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2answers
30 views

Prove Kolmogorov's SLLN by martingale.

Suppose $\xi_i$ are i.i.d. and $\mathbb E(|\xi_1|)\lt\infty$ Let $X_n=\sum_{i=1}^n\xi_i$ Then we have $\frac{X_n}{n}\to \mathbb E(|\xi_1|) $a.s. In the proof of this theorem: ...
1
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0answers
13 views

Simple random walk conditioning on non-return

Consider a simple symmetric random walk on $\mathbb{Z}$, $(S_t)_{t \geq 0}$, with $S_0=0$. Let $P_{k,j}$ be the probability that the walker hits the point $k$ without returning to the origin in ...
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0answers
32 views

What is meant by Stable Law?

I am reading a very complex paper consider a set of random variables $\left\{ X_{i}\right\} _{i=1}^{\infty }$ whose common distribution $F_{X}$ belong to the domain of attraction of an $\alpha ...
2
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0answers
37 views

Almost sure limit of $\log(X_1 + X_2 + … + X_n) - \log(n)$

Let $X_n$ be an i.i.d. sequence of positive random variables with expectation 2 and variance 1. What is the almost sure limit of $$\log(X_1 + X_2 + ... + X_n) - \log(n)$$ as $n \to \infty$ Would it ...
0
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1answer
45 views

Expectation of sum of $n$ random variables.

Let $X_1, X_2, X_3 \ldots, X_n$ be n random variables which take values from $\{+1,-1\}$ uniformly. Let $S_n$ be defined as $$S_n =|X_1+ X_2+ X_3+ \dots +X_n| $$ Find the expectation $E(S_n)$ of the ...
0
votes
1answer
55 views

The probability of having $k$ successes before $r$ failures in a sequence of independent Bernoulli trials

Problem Find the probability of having $k$ successes before $r$ failures in a sequence of independent Bernoulli trials with $p$ being the probability of success. I thought of using the Binomial ...
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2answers
15 views

Geometric and binomial distribution problem

Let $X \sim Bi(n,p)$, and $Y \sim \mathcal{G}(p)$. (a) Show that $P(X=0)=P(Y>n)$. (b) Find the number of kids a marriage should have so as the probability of having at least one boy is $\geq ...
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1answer
50 views

The probability of hitting a target at least twice, conditioned on hitting it at least once

I am working on some exercises related to random variables, since the subject is new to me, I feel a little insecure about my answers, I'll write the problem I've worked on to check if my solution is ...
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0answers
25 views

A R.V. is approximated as normal distribution by CLT. Can I use CLT again if I want to sum it with another R.V.?

My problem is, I have many independent normal random variables and since I need to have their squared distribution, I use Central Limit Theory to approximate the sum of its squared distribution as ...
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1answer
39 views

Is $\exp(-2\sin^2t)$ a characteristic function?

Is $\exp(-2\sin^2t)$ the characteristic function of some random variable?
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2answers
33 views

Deriving exponential distribution from geometric

Let $\lambda$ be the expected number of events in a unit time interval $[s,s+1]$ (events are independent of each other and of the time interval), and $T$ a continuous random variable that represents ...
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2answers
46 views

Showing that $\mathbb{E}|X\ln X| < \infty$ and $\mathbb{E}Z = \mathbb{E} X \ln X$ for given PDF of $Z$.

$X$ is real random variable such that $\mathbb{P}(X > 0) = 1$, $\mathbb{E}X^2 < \infty$, $\mathbb{E}X=1$. Let $Z$ be real random variable such that $\mathbb{P}(Z \in ...
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votes
3answers
29 views

Variance of the sum of sample means

Let $X$ be a random variable with normal distribution with mean $ \theta$ and variance $ a>0$. Let $ Y $ be a random, variable with normal distribution with mean $\theta$ and variance $b>0$. ...
0
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0answers
16 views

Generate Correlated Normal and Log-Normal Random Variable

The standard approach for generating two normally distributed random variables some with correlation $\rho$ is explained here: Generate Correlated Normal Random Variables. Now let $X,Y$ be normally ...
1
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1answer
18 views

Bound on variance of random process when signal is known

I am reading this paper (link to a Nature paper, may not be accessible) and I encountered the following. I have very little experience in probability theory and I could not find much helpful in ...
3
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1answer
49 views

Convergence of discrete random variables, show $\frac{S_n}{\sqrt{n}}\to0$ a.s.

Let $X_n$ be a sequence of independent discrete real random variables, with discrete density $$p_{X_n}(x):=\Pr(X_n=x)= \cases{ 1-\frac1{n^2} & \text{if } x= 0\cr \frac1{2n^2} & \text{if ...
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1answer
26 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 ...
0
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1answer
33 views

Is the set $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ equal to $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$?

Difference between $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ and $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ are the sets equal ? I think they would ...
0
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1answer
58 views

What is the mean and variance of $Y$, where $Y$ is sum of iid's

Here's my work for part a. I could use clarification on part b and d. Is part d the same as part a ($E[A_n] = E[Y]$) ? a) $$E[Y_n] = E[\frac{X_n}{2^n}]$$ ($X$'s are iid so...) $$= \frac{E[X]}{2^n} ...
2
votes
1answer
18 views

Why $E[X|\mathcal{G}]=X$ if $X$ is $\mathcal{G}$-measurable?

If $X$ is a $\mathcal{G}$-measurable random variable, why $E[X|\mathcal{G}] = X$? I know the intuition (basicly we're conditioning on the same informations on which $X$ is defined, $\sigma(X)$, we ...
0
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2answers
56 views

For random variables, show that $\limsup\limits_nX_n<\infty\Longrightarrow \sup_n X_n<\infty$

Why is the following true ? $$\limsup\limits_nX_n<\infty\Longrightarrow \sup_n X_n<\infty$$ where, $X_n's$ are random variables. If we consider only finitely many $X_n$, say ...
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2answers
47 views

Compensation Question

I want to create a compensation system which takes into account two variables. Lets say I have $1M to distribute among ten employees who produce widgets. I want to compensate each employee by two ...
0
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1answer
40 views

Convergence sequence of random variables

I have this problem about a sequence of normals. $(X_n)_{n\geq 0}$ is defined as $$X_{n+1}=aX_n+U_{n+1}$$ $X_0=0$, where $(U_n)_{n\geq1}$ is a sequence of i.i.d random variable normally distributed ...
2
votes
1answer
76 views

Mean of Piecewise function resting on IID random variables

Suppose IID random variables $X_t \sim X$ with support on $[0,1]$ and continuous CDF $F(\cdot)$. I wish to compute the expected value (mean) of the a piecewise function with form $$ \Phi (x,\mu) = ...
3
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3answers
51 views

Transformation(?) of Random Variables

There are two independent Gaussian R.Vs: $U:N(-1,1)$ and $V:N(1,1)$ How do I go about finding the PDF of the following transformations? X = U+V T = (U+2V, U-2V) W = U (with 50% chance), V (with ...
2
votes
1answer
13 views

Distribution of a function of a normally distributed variable

Let's say you have a random variable $X$, which is normally distributed according to $X \sim \mathcal{N}(1,2)$. With $1$ being the mean and $2$ being the variance. Now let's say that there is another ...
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0answers
23 views

Hint for KKT Optimization problem

Can anyone help me with the following optimization problem please? I have to find the $\max f(c,y_1^1,\cdots,y_{N-1}^1,\cdots,y_1^M,\cdots,y_{N-1}^M)=c$ subject to the constraints ...
3
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2answers
81 views

How to give rigorous proofs of these two limit statements?

Let $X$ be a random variable with cumulative distribution function $F(x)$. Then how to rigorously prove the following two limit statements? $\lim_{x \to - \infty} F(x) = 0$. $\lim_{x \to + \infty} ...
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1answer
35 views

Mean of max vs max of mean

If I have say an $n$ collection of 10 random variables $X_1, \ldots, X_{10}$ (so an $n \times 10$ matrix of values) from some underlying distribution whether Gaussian or uniform, and I calculate ...
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1answer
42 views

Prove that the series $\sum\limits_{n=0}^{\infty}X_n$ converges almost surely

I'm trying to solve the following Problem: Let $(X_n)_{n\ge 1}$ be a sequence of real valued random variables defined on some probability space $(\Omega, \mathcal{A},P)$. Assume that there ...
1
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1answer
19 views

Does negative part of a standardized random variable converge to negative part of a $\mathcal{N}(0,1)$?

I know how to prove that any standardized random variable converge in distribution to a $\mathcal{N}(0,1)$, I was wondering if even $f((S_n-n)/ \sqrt{n}))$ converge to $f(\mathcal{N}(0,1))$, in ...
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0answers
33 views

About independent random variables

Suppose that $\{X_n\}_{n\in\mathbb N}$ are identical distribued and independent random variables with values in $\mathbb Z$. I don't understand why ...
5
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0answers
52 views

Variational formulations in group theory?

I apologise if this is a naïve question. Are there any known / widely applicable / important variational formulations in (finite) group theory? That is, a relationship of the form $$\alpha(G) = ...
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0answers
25 views

Testing for the independence of random variables

In probability theory, $X$ and $Y$ are independent if: $f_{X|Y}(x|y)=f_X(x)f_Y(y)$ If I have sample $Y_1,...,Y_n$ and I would like to test if $Y_i$ is independent from the rest of the sample, I ...
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2answers
33 views

Identifying the distribution which represents a negative binomial distribution as a compound poisson distribution

Suppose that the random variable $X$, which has a negative binomial distribution with probability $p$ and parameter $r$, can be represented as the summation of $N$ iid random variables $Y_1, Y_2, ...
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0answers
34 views

Probability distribution of k consecutive successes with n maximum trials

Let $X$ be a random variable that represents the number of trials of a given experiment. The outcome of a single trial is a Bernoulli random variable, with probability of success $p$, and trials are ...
0
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1answer
43 views

Expected value and variance of random process

Let $U,V$ be random variables with distributions $\mathcal{U}(-1,1)$ ,$\mathcal{E}(2)$ (uniform and exponential). If $U$ and $V$ are independent what is the variance and expectation of the random ...