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Questions tagged [random-variables]

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

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1answer
17 views

Difference of normal random variables

I have two random variables $$X_{s+t} \sim N(0, s+t)$$ $$X_s \sim N(0, s)$$ where $s \leq t$. How do I show that... $$X_{s+t} - X_s \sim N\left(0,s + t + s -2\sqrt{s(s+t)} \right)??$$ I understand ...
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0answers
16 views

Quadratic variation of sum od random variables

Let $N = (Nt)_{t>0}$ be a Poisson process and consider random variables $Z_n$, $n \in N$. Compute the quadratic variation $[X]_t$ where $X_t =\sum_{n=1}^{N_t} Z_n$. What I did was plugging $X_t$ ...
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1answer
30 views

Joint PDF transform using jacobian

Seriously I dont have any idea what is this thing called. I know how to find Joint PDf of two variables.. But i dont know how to transform it in other variables ? Do they require Jacobians? Here is ...
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1answer
35 views

Find two random variables X and Y such that P(X<Y)=2/3

The problem is in the title (sorry if that's not kosher) but I have absolutely no clue how to get started. What would be a good approach to take with this one? I don't believe I have even seen a $\Bbb ...
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2answers
31 views

How to use the law of total variance

I know that the law of total variance states $$Var(X)=\Bbb E[Var(X|Y)]+Var(\Bbb E[X|Y])$$ But how does one treat $Var(X|Y)$ and $\Bbb E[X|Y]$ as random variables? For example, say we know that $$\Bbb ...
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1answer
23 views

Finding expected value when conditional distribution is known

If the distribution of $Y$ conditional on $X=x$ is known, and the distribution of $X$ is known, what would be the general process for finding the expected value $\Bbb E[Y]$? IS there a general ...
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0answers
29 views

Is random variable a way to search for actual input numbers?

The formal definition of a random variable is Random variable over a sample space is a function from sample space to $R$ I want to get intuitively what actually it is doing. In this context, I ...
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2answers
77 views

What is the formalism that allows Random Variables to be treated algebraically like real or complex numbers?

We all know that if we have a variable x, then there is a meaning to - for example - $$y=e^x$$. And we all know how to manipulate that algebraically and to do calculus. For example, if $$y_1=e^{...
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1answer
29 views

Is the sum of two (non-real) random variables necessarily a random variable?

Please note that I'm working with the following definition of random variable, which allows for a codomain other than $\mathbb{R}$. Definition: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a ...
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0answers
22 views

Proving expectation and variance of a function of a random variable tends to a fix point

Given $f:\mathcal{X} \rightarrow \mathbb{R}$ is a continuous function and $\mathbb{E}_{Q(X)}[X] \rightarrow x^\star$ ($x^\star$ is a fix number), $\mathbb{V}\text{ar}_{Q(X)}[X] \rightarrow 0$. How can ...
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0answers
23 views

Bound for the expectation of a function of a Gaussian random vector

A paper I am currently reading seem to be using the bound $$ E_h\left[\inf_{\|z\|_{\infty}\leq t} \|z-h\|^2\right] \leq n\frac{1}{\pi}\frac1t e^{-t^2/2},$$ where $n$ is the dimension of the vector, ...
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0answers
14 views

Proving convergence of expectation and variance given Rényi's $\alpha$-divergence tends to 0

I denote $p, q$ as density function of $P, Q$. Given $Y, X$ are random variables and \begin{align} \int q(x)\mathbb{D}_{\alpha}[p(Y\mid X=x)\,||\,p(Y\mid x^\star)] \,dx \rightarrow 0 \end{align} ...
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0answers
13 views

range of correlation of the linear combination of two random variables

$X_1, X_2, Y$ are normal random variables, If we know $\operatorname{corr}(Y, X_1) = \rho_1$, $\operatorname{corr}(Y, X_2) = \rho_2$, $\operatorname{corr}(X_1, X_2) = \rho_3$ What's the range of the ...
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3answers
85 views

Why is $y$ separated into two intervals?

So, here's a question and a solution to part b). I do not understand why they make $y^{1/2}$ belong to interval $[0,1)$ and then separately to the interval $[1,3)$.
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0answers
21 views

Joint PDF of two dependent exponential random variables [on hold]

I need help in following Let's assume we have two identically distributed exponential random variables $X$ and $Y$. What will be the joint pdf of those random variables, i need to consider two cases, ...
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2answers
36 views

How to find the mean and variance of minimum of two dependent random variables?

Assume that $X$, $Y$, and $Z$ are identical independent Gaussian random variables. I'd like to compute the mean and variance of $S=\min\{P, Q\}$, where : $$Q=(X-Y)^2$$ and $$P=(X-Z)^2$$ Any help is ...
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0answers
21 views

For which random variable is the spectral density a PDF?

The spectral density of a stationary random process is the inverse Fourier transform of the autocorrelation function of that random process. This spectral density is a probability density function for ...
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0answers
19 views

Example of a pair of random variables

I have this example of random variables let $ \ \ \ f_{XY}( x,y) = \begin{cases} cx \ \ \ \ 0 \leq x \leq 1 \ , \ 0 \leq y \leq 1 \\0 \ \ \ \ \ other \end{cases} $ a) determine the $c$ ...
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1answer
19 views

Distance of squared random variables and upper bound

I got two sequences of random variables, $(X_n)_n$ and $(Y_n)_n$, and I know that $| X_n - Y_n | \leq C a_n $, for some constant $C$ or equivalently $|X_n - Y_n | = \mathcal{O}(a_n)$. Now I want to ...
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0answers
7 views

Point is randomly selected from area, find the mean of abscissa.

my question sounds like this: Point is randomly selected from area, which is limited with parabola y = x2 and straight lines y = 0, x = 0.97, x = 2.78. Find this point the mean of abscissa. This ...
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0answers
39 views

Expectation of an inner product in an infinite dimensional Hilbert space

Let $\mathcal{H}$ be a Hilbert space with the Borel $\sigma$-algebra. Let $(\Omega, \mathcal{F}, P)$ be a probability space and $x,y$ two $\mathcal{H}$-valued random variables, i.e. measurable maps ...
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1answer
25 views

Finding the estimator of π(1-π) from a random sample of n Bernoulli trials.

A random sample of n independent Bernoulli trials with success probability π results in R successes. Derive an unbiased estimator of π (1 − π). So, from what I understand (correct me if anything I ...
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1answer
29 views

Finding density function of random variable

Choose an uniformly distributed random variable $U$ on the unit interval $[0,1]$. Then, what is the probability density function of $Y= \ln(U+ 1)$? I know the density function is the derivative of ...
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1answer
43 views

Transforming sum of n exponential distribution to a Poisson distribution

Let $X_1,...,X_n$ be i.i.d exponential random variable with mean $\lambda$ $S=X_1+...+X_n$ So by finding the mgf of S, we get that $S \sim \operatorname{Gamma}(n,\lambda)$ The problem I am stuck ...
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1answer
12 views

Find mean and variance from mgf where t is denominator

For continuous random variable X, pdf: $f_{X}(x)=2(1-x), x\in[0,1]$ mgf: $M_{X}(t)=\frac{2(e^t-t-1)}{t^2}$ Problem is to find mean and variance from mgf, I tried using $\frac{d}{dt}M_{X}(0)$ and $\...
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2answers
17 views

What is the average number of matches when randomly picking letters

Suppose we have three pieces of paper. On the first one you have the letter A, on the second on the letter B, and on the third one the letter C. Now suppose I'm going to randomly pick each one from a ...
2
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1answer
54 views

Suppose $X_1, \dots, X_n, Y$ are independent random variables. Prove that $X = (X_1, \dots, X_n)$ and $Y$ are independent variables.

Suppose $X_1, \dots, X_n, Y$ are independent random variables. Prove that $X = (X_1, \dots, X_n)$ and $Y$ are independent variables. My attempt: Fix $A \in \mathcal{R}$ (a Borel subset of the real ...
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0answers
22 views

Lognormal distributed random variable excercise

Let Y be a random variable distributed Lognormally, Q1 be it's first quartile, Q3 it's third quartile and M be its median: prove that- M - Q1 < Q3 - M I have managed to figure out that Q3 > M > ...
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0answers
14 views

Convergence on Geometric distribution [on hold]

Suppose that $Xn$ ∼ $Geo( {λ/n+λ} )$ with $n = 1,2,...$ where λ is a positive constant. Show that $Xn/n$ converges on when n → ∞, and determine the parameter of the limit distribution.
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1answer
37 views

Expected value in a linear combination

I have a random variable Y, that is defined by: $$Y = aX_1 + bX_2$$ Where we know $X_1 $ and $X_2$ are independent. How do I write out $EX_1$ and $EX_2$ in terms of only a, b, EY, and VarY? I ...
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1answer
49 views

Inserting random numbers from 1 to $n^2$ in a matrix of size $n \times n$

I have two matrices of size nxn with random numbers that are in range of $1$ to $n^2$. I'm trying to calculate the probability of : the numbers 1 and 9 are present in the same indices in the two ...
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0answers
8 views

Approximation of mean of a rational function of random variables

Let $\xi_i$ with $i\in\{1,\dots,n\}$ be iid random variables and let $Q(x,y)$ be a rational function. I need to compute one $x$ that satisfies $$\frac{1}{n}\sum_{i=1}^n Q(x,\xi_i)=0.$$ This is a ...
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0answers
19 views

binomial distribution problem prove or disprove

Let $Y_n\sim B(10, 2p) \wedge 0<p<\frac{1}{20}$. prove or disprove: $\lim_{n\rightarrow\infty}p(|Y_n-20p|>\epsilon) = 0$ my try: $p(|Y_n-20p|>ϵ)=p(Y_n-20p>ϵ)+ p(Y_n-20p≤-ϵ) =p(Y_n&...
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1answer
14 views

Existence of random points

Let $\mathbb{S}^k$ be the $k$-dimensional unit sphere and let $\sigma(\mathbb{S}^k)$ be its surface area. Suppose we have a regular area partition $\{S_i\}_{i=1}^n$ of $\mathbb{S}^k$ with constant $c&...
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0answers
42 views

Understanding i.i.d. random variables from product measure space perspective

I have a very weak background in measure theory, and I am having some troubles understanding i.i.d. random variables from a measure theoretic perspective. Let $X$ be a random variable defined on a ...
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1answer
31 views

$X$, $Y$ i.i.d r.v's. Prove that $\mathbb{E}[X\mathbb{1}_{\{X+Y \in B\}}] = \mathbb{E}[Y\mathbb{1}_{\{X+Y \in B\}}] $

Let $X, Y$ be i.i.d random variables with finite expected values. I want to justify that $$ \int_{\{x+y \in B\}}x\mu(dx)\mu(du)=\int_{\{x+y \in B\}}y\mu(dx)\mu(du). $$ I would appreciate any hints, ...
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2answers
73 views

Expectation of dependent Bernoulli sum

I want to estimate the expected value of the following sum of random variables, $$ Y = \sum_{i=1}^N X_i $$ where each $X_i$ is a Bernoulli random variable. In particular, $$ X_1 = \begin{cases} 1, &...
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0answers
24 views

Find variance D[ξ-2η] when ξ and η are independent

My question is: Random variables ξ and η are independent and with with equal probabilities acquires values -3, 0, -3. Find variance D[ξ-2η] I've tried to solve like this: But I'm not sure if ...
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1answer
31 views

Does $|X_n|\le\Delta_n+\delta$, with $\Delta_n\overset{p}\to0$ imply $|X_n|\overset{p}\to0$?

Suppose $\Delta_n\overset{p}\to0$, and for any $\delta>0$, we have $$|X_n|\le\Delta_n+\delta.$$ Can we conclude that $$|X_n|\overset{p}\to0?$$ Here $\Delta_n\overset{p}\to0$ means that for any $\...
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1answer
51 views

Deriving distribution from conditional distribution

Hi guys I am having problems deriving $P(X = k)$ if $P(X = k|X+Y = n)$ = ${n}\choose{k}$ $\times$ $2^{-n} $ X and Y are i.i.d. random variables with values in $\mathbb{N_0}$. After playing a bit ...
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1answer
37 views

Statement about convergence in probability

If $X_1, X_2, ..., X_n, ...$ are same distributed and independent random variables and $n \mathbb P (|X_1| > n) \to 0$, $a_n = \mathbb E\left[X_1I_{|X_1| < n}\right]$, does this imply that $X_1+...
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1answer
38 views

Random variables with Gamma distribution and convergence in probability.

The problem states that $\alpha>0$ and for each $n\in\mathbb{N},\ X_n:\Omega \rightarrow \mathbb{R}$ is a random variable on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with the gamma ...
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0answers
30 views

Asymptotic distance between distributions in delta method

Let $X_n$ be a sequence of real valued random variables such that $a_n(X_n - \mu) \xrightarrow{\text{d}} X$ for $a_n \to \infty$. Delta method states that $a_n(g(X_n) - g(\mu)) \xrightarrow{\text{d}} ...
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1answer
26 views

Does $\xi_{ni}\overset{p}\to0$ imply $\frac1n\sum_{i=1}^n\xi_{ni}\overset{p}\to0?$

Suppose for each $i=1,\cdots,n$, we have $$\xi_{ni}\overset{p}\to0.$$ Can we claim that $$\frac1n\sum_{i=1}^n\xi_{ni}\overset{p}\to0?$$ Here $\overset{p}\to$ means convergence in probability. That ...
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0answers
18 views

Multistage random variable

First we sample from a probability space $\Omega$ and observe the value of $X$, then we sample $Y$ from a probability space $\Omega_X$. In terms of measure theory, what is $Y$? Let $X: \Omega \to I$ ...
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1answer
21 views

About the definition of independent random vectors

I just have a question concerning the definition of independent random vectors. The random vectors $X=[X_1, \ldots, X_n]$ and $Y=[Y_1,\ldots, Y_m]$ are independent means that $X_1,...,X_n,Y_1,...,Y_m$ ...
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0answers
22 views

stochastically independence

I never took stochastic courses and need a proof for this task to continue my work at another problem. Can somebody help me out? Let $X$ and $Y$ be stochastically independent real discrete random ...
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1answer
21 views

How to efficiently generate random points in a difference of two disks?

Consider two disks: $S$ with radius $R_S$ and position $(x_S,y_S)$, and $M$ with radius $R_M$ and position $(x_M,y_M)$. Let's denote the difference between these two sets as $$V=S\setminus M.$$ Now ...
0
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1answer
32 views

Expected value of multiplication of matrices

While reading through Xu et al. (2016) I stumbled upon this proof: $$ \begin{align} \mathbb{E}[Wyy^\intercal W^\intercal] & = \mathbb{E}[Wyy^\intercal W^\intercal-W\mathbb{E}[y]\mathbb{E}[y]^\...
1
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1answer
50 views

The probability of two samples of a waveform to be separated by certain amount

I am trying to mathematically model the probability of observing a waveform crossing based on two of its samples. Im particular, I want to know the effect of the sampling frequency and phase on this. ...