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Questions tagged [random-variable]

A random variable or stochastic variable is a value that is subject to chance variation (i.e., randomness in a mathematical sense).

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A random variable with a step function CDF is discrete?

Consider the step function $$ \Delta(x;\lambda,\mu)\equiv \sum_{j=1}^J \lambda_j\times 1\{\mu_j\leq x\} $$ where $\lambda_j\geq 0$ $\forall j$; $\sum_{j=1}^J \lambda_j=1$ $\mu_j\in \mathbb{R}$ $\...
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Central limit theorem doubt [duplicate]

Consider a random variable $X$ that takes only positive values, (in my case the r.v. $X=Y^2$ where $Y$ is a random variable itself). We know that from the CLT we have that $\sqrt{n}(\bar{x}-\mu)\...
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Is non-central chi-square distribution sub-exponential

For $X \sim \mathcal{N}(0,1)$, and $a > 0$, is $(a+X)^2$ sub-exponential? After some algebra, I get $$\mathbb{E}\left[ e^{s(X-\mu_X)} \right]= \frac{1}{\sqrt{1-2s}} \exp\left( -s \frac{1-2s(a^2+1)}{...
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Sum of Discrete Random Variables [duplicate]

If I have two independent discrete random variables, say, $$ X \in \{1,3,10,20\} $$ and $$ Y \in \{2,3,5,9,11,15\} $$ and let $$Z = X + Y $$ be the sum of two variables. Also, each value taken by ...
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Distribution of a square of a random variable [duplicate]

If X is normally distributed with mean mu and variance sigma^2, then what can I know about the distribution of X^2 ?
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1answer
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joint PMF in a class of n students

A class of n students takes a test in which each student gets an A with probability p, a B with probability q, and a grade below B with probability 1 − p − q, independently of any other student. If X ...
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1answer
42 views

PMF and independence with two discrete random variables?

Each of n people (whom we label 1, 2, . . . , n) are randomly and independently assigned a number from the set {1, 2, 3, . . . , 365} according to the uniform distribution. We will call this number ...
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2answers
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Conditional expectation function

Consider the standard linear regression model given by $Y = XB + \varepsilon$. $E[Y\mid X] = XB$ if $E[\varepsilon \mid X] = 0$. We say that the conditional expectation function is a random ...
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Quantiles of transformed series

Possibly stupid question, but I need to ask. Let's have a time series $S_{n}$ of a same market asset. Let's $R_n = ln(S_n/S_{n-1})$ be an asset returns. So, I could forecast same $\overline{R}_{N+1}, \...
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1answer
29 views

Random variables - proof of convergence in probability

I've got this exercise from lecture notes, but I couldn't find an answer. For each positive integer $n$, let $X_{n}$ be a non-negative random variable with $\mathbb{E}[X_{n}] < \infty$. Prove that ...
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Summation Bounds When Finding Transformation of 2 Poisson Random Variables

I am reviewing some material on functions of several random variables from Section 7.4 of John E. Freund's Mathematical Statistics, 6th Edition, and I'm stumped on how the author gets the upper bound ...
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How to find conditional expectation$(X_1+X_2)^2$ given $X_1 = X_2$?

How do I show that $E[(X_1 + X_2)^2|X_1=X_2] = 2\sigma^2 + 4\mu^2$. When $X_1$ and $X_2$ follows $N(\mu,\sigma^2)$ independently. As $X_1 = X_2$ is given, then I suppose I only need to find $E[4{X_1}^...
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1answer
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Expected time to visit all countries by random flight paths [closed]

Say there are $n$ different countries, the flight starts from some initial country. At each step, the flight can go to a random country other than the one where it currently is. The probability of ...
3
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1answer
31 views

Sum of square of gaussian random variable and exponential random variable

I have 2 independent RVs $s$ and $N$ with distribution as below: $\begin{array} { c } { f _ { s } ( s ) = \frac { 1 } { \sqrt { 2 \pi \sigma ^ { 2 } } } e ^ { - s ^ { 2 } / 2 \sigma ^ { 2 } } } \\ { \...
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22 views

Calculating probability of two random variables [duplicate]

I'm having trouble calculating the probability of two random variables. A question on an exam review is P(X+Y>38). I know how to do this problem with just one variable, using z scores, but i'm not ...
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2answers
45 views

Symmetrical nature of binomial distribution [closed]

If $X$ is a $binomial(n,p)$ random variable, and $P(X<15)<0.5$ Find $n$. Less than $30$ More than $30$ Equal to $30$ None of the above. I know that binomial distribution is symmetric when ...
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1answer
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Interpreting sample space of a unknown random variable

After looking for a dozen of questions on cross validated, I've decided to write my own. We know the following mapping is called a random variable $$X:\Omega\to \mathbb{R}$$ where $\Omega$ is set of ...
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2answers
37 views

Two distributions, same mean, different variance: Stochastic dominance for deviation from mean?

Say you have two (bounded) random variables, $X$ and $Y$, on the same discrete probability space, such that $E(X)=E(Y)$ but $Var(X) < Var(Y)$. Do I know that, for any $k \geq 0$, $$ \text{Prob}(|X-...
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0answers
32 views

Mean and Variance of $Z=(X^2+Y^2)^{0.5}$ [closed]

I want to find the mean and variance of $Z=(X^2+Y^2)^{0.5}$ where $X$ and $Y$ are independent with mean $0$ and $Var(X)$ and $Var(Y)$ are not equal.
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Bounds on quantiles of the minimum of summations of (possibly dependent) random variables

Suppose I have $2N$ continuous random variables $X_1, \ldots, X_N, Y_1, \ldots, Y_N$ and that I can evaluate the quantiles of the respective distributions. Given a value $w \in [0, 1]$ I would like to ...
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2answers
28 views

Normal distributed random variables with constraint?

Consider $n$ random variables $X_i$ with $i=1,2,...,n$, each drawing values from identical normal distributions with mean $\mu=0$ and standard deviation $\sigma=const.$ so that expectation values are $...
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$\mathbb{E}[\sigma(r)^2]$ with $r \sim \mathcal{N}(0,1)$

Start with a random variable $r \sim \mathcal{N}(0,1)$. Now consider the random variable $\sigma(r)$ formed by passing it through a standard logistic function $\sigma(x) = \frac{1}{1 + e^{-x}}$. I ...
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1answer
71 views

What does it mean to generate a random variable from a distribution when random variable is a function?

I am looking at a reference for sampling from a distribution, and the first step of the so-called algorithm states:http://www.columbia.edu/~ks20/4703-Sigman/4703-07-Notes-ARM.pdf Generate a random ...
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1answer
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probabilistic distribution of a variable which are based on random imputs

In practice, I have a variable x, which is based on (b,c,d). We may have a physics based math formula to describe the relationship between x and (b,c d), i.e., x=f(b,c,d). Beforehand, we may know the ...
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Mutual info between continuous and discrete variables from numerical data

I am looking for references/measures to estimate the mutual information between a continuous (C) and discrete (D) variable, given a real-world (i.e. finite sample) data set. C is uniformly distributed ...
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1answer
21 views

Probability that one random variable using the Beta Distribution being greater than another, bounded intervals

I am doing some practice problems to prepare for my statistics exam, and I just want to know if my reasoning is correct on one problem, and if not, I want to know how I should reason through this. The ...
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1answer
16 views

Covariance of sums of pairs of correlated variables

Take two vectors of normally-distributed random variables $\mathbf{x} = (x_1, x_2, \ldots x_n)$ $\mathbf{y} = (y_1, y_2, \ldots y_n)$ where the covariance of each pair $(x_i, y_i)$ is known, $\...
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Questions on Notation for PMF and Expectation

Before diving into the Stanford CS229 Machine Learning notes online, I decided to go through the course's notes on probability review and had a few questions. In section 2.2, it states A ...
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1answer
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Joint cumulative distribution of independent random variables

X,Y,Z are non negative random variables which are independent and uniformly distributed in [0,1] and let $\alpha$ be a given number in [0.1]. Now how to compute $\text{Pr}(X+Y+Z>\alpha \;\;\; \&...
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1answer
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Continous random variable short proof

I am given the problem: If X is a continuous random variable with cumulative distribution function F and density function f, show that the random variable Y = X^2 is also continuous and express its ...
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1answer
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Which random effects to include in a mixed effects model?

I am analyzing data from a perceptual decision making experiment (10 participants, 1800 trials each). Participants made perceptual decisions (3 possible responses) and then rated their confidence on a ...
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2answers
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What are the predictor variables in a neural network?

In a linear regression model, the predictor or independent (random) variables or regressors are often denoted by $X$. The related Wikipedia article does, IMHO, a good job at introducing linear ...
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Mutual info between discrete and continuous RV with history dependence

I am looking for literature references/measures to compute the mutual information between a continuous variable (time series) and a binary variable (a temporal sequence of 0's/1's). Briefly, a time ...
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2answers
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Suppose $\mathbf{X, Y}$ are independent random vectors. Are their components independent? [duplicate]

Let $\mathbf{X} = (X_1, \dots, X_p)^\top$ and $\mathbf{Y} = (Y_1, \dots, Y_p)^\top$ be independent. Does it then follow that $X_i$ is independent with $Y_j$ i.e. cov$(X_i, Y_j) = 0$?
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1answer
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How to generate samples of Poisson-Lognormal distribution

I would like to compute samples of the number of product purchased in a supermarket. I want to model it with a mixed Poisson lognormal distribution. Items purchased $x$ of a given consumer follow a ...
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2answers
46 views

Rate of convergence of sum of two random variables

Let $X_n$ and $Y_n$ be random variables such that $X_n=o_p(1)$, $Y_n=o_p(1)$, $X_n - Y_n = o_p(1)$. Is the following correct? $o_p(X_n) + o_p(Y_n) = o_p(|X_n - Y_n|)$
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Generating correlated random numbers from uncorrelated ones

As per this paper - https://www.nag.com/IndustryArticles/fixing-a-broken-correlation-matrix.pdf -- it is written: I am not getting sure show to create the third variable X3 and how it transforms into ...
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1answer
35 views

What is the probability that at least three guilty parties are caught at the same time and at least four of the innocent are released?

A lie detector will be used by police to investigate 10 suspects of involvement in a particular crime. Admit that among them, five are guilty (but will plead innocence) and the other five are really ...
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0answers
21 views

Continuous Stochastic Processes examples

I am trying to understand various types of stochastic processes. In order for that to happen, I needed some simple examples to be built so that I can build an intuition about them. According to the ...
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1answer
35 views

Multi-dimensional CDF on a discrete support

Suppose I have two discrete-support random variables, $X$ and $Y$. They have joint CDF $F(X,Y)$. If I want to find $\Pr(a \leq X \leq b , c \leq Y \leq d)$. It is obviously not: $F(b ,d)-F(a-1 ,...
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0answers
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Discrete Stochastic Processes examples

I am trying to understand various types of stochastic processes. In order for that to happen, I needed some simple examples to be built so that I can build an intuition about them. According to the ...
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0answers
30 views

Some simple examples to understand various types of Stochastic Processes

I am trying to understand various types of stochastic processes through some analogical examples using simple experiments like a coin toss or die roll. The book of Hwei Hsu (Chapter-5, Page-162-165, "...
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1answer
46 views

Expected value of ratio of two function of the same random variable

Let $X$ be a r.v. with absolutely continuous distribution and continuous strictly positive density $f: \mathbb{R} \rightarrow [0, \infty)$ and let $g$ a further given continuous density function. Set ...
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Importance of the right-continuity of filtration in definition of strong Markov Property

Taking the definition from wikipedia, With $X = (X_t : t \geq 0) $ as a stochastic process on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with natural filtration $\{ \mathcal{F}(t) \}_{t \...
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1answer
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Expectation of product - formula check

if $x_i$ are IID continuous random variables, with $E[x_i]=μ$, is the following correct? $E\left[\prod_{i=1}^n(1+⍺\space x_i)^i\right] = \prod_{i=1}^nE\left[(1+⍺\space x_i)^i\right] = \prod_{i=1}^...
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0answers
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Why is it necessary to assume that a sample vector consists of n sample variables instead of assuming that we have a sample of size n?

This is a very basic question, but I need help to grasp the concept. From what I understand: If I carry out a survey on body weight, every single answer to the question "How much do you weigh?" is a ...
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1answer
38 views

Negative variance, what is wrong?

I am trying to obtain the variance of a function of two random variables $$f(\boldsymbol x):= x_A (e^{k(x_A+x_B)}-1)$$ where $\boldsymbol x = [x_A, x_B]^T$. Additionally, I know that $\operatorname{...
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3answers
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What does it mean to say that $X_1, X_2$ have a “common” Normal distribution?

An exercise question asks Let $X_1, X_2$ be rvs having a common Normal distribution $N(0,1)$ with $\operatorname{Corr}(X_1, X_2) = \rho$. Calculate the coefficient of upper tail-dependence for all $...
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0answers
40 views

Understanding sum of square deviations [duplicate]

Given $X_1...X_n\stackrel{iid}{\sim} N(\mu,\sigma^2)$ and $U=\sum_{i=1}^n (X_i-\overline{X})^2$, why is $U\sim\sigma^2 \chi_{n-1}^2$ ? And what would be the distribution of $V=\sum_{i=1}^n (X_i-\mu)^...
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Finding Chernoff bounds maximum estimators

I am currently trying to resolve the following exercise about Chernoff bounds: Let $X_{1}, X_{2}, \dots, X_{n}$ be independent, identically distributed (i.i.d) random variables with distribution $N(0,...