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).
3
votes
1answer
38 views
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
$\lambda_j\geq 0$ $\forall j$; $\sum_{j=1}^J \lambda_j=1$
$\mu_j\in \mathbb{R}$ $\...
0
votes
0answers
16 views
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)\...
-1
votes
0answers
16 views
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
$$\mathbb{E}\left[ e^{s(X-\mu_X)} \right]= \frac{1}{\sqrt{1-2s}} \exp\left( -s \frac{1-2s(a^2+1)}{...
2
votes
0answers
26 views
Sum of Discrete Random Variables [duplicate]
If I have two independent discrete random variables, say,
$$ 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 ...
1
vote
0answers
8 views
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 ?
0
votes
1answer
25 views
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 ...
1
vote
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 ...
2
votes
2answers
46 views
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 ...
0
votes
0answers
22 views
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}, \...
1
vote
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 ...
0
votes
1answer
13 views
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 ...
0
votes
0answers
23 views
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}^...
3
votes
1answer
15 views
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
votes
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 } } } \\ { \...
0
votes
0answers
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 ...
1
vote
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 ...
0
votes
1answer
48 views
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}$$
where $\Omega$ is set of ...
1
vote
2answers
37 views
Two distributions, same mean, different variance: Stochastic dominance for deviation from mean?
Say you have two (bounded) random variables, $X$ and $Y$, on the same discrete probability space, such that $E(X)=E(Y)$ but $Var(X) < Var(Y)$. Do I know that, for any $k \geq 0$,
$$
\text{Prob}(|X-...
1
vote
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.
1
vote
0answers
18 views
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 ...
1
vote
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 $...
3
votes
0answers
52 views
$\mathbb{E}[\sigma(r)^2]$ 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 ...
2
votes
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 ...
0
votes
1answer
17 views
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 ...
0
votes
0answers
19 views
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 ...
1
vote
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 ...
0
votes
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,
$\...
1
vote
0answers
16 views
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 ...
2
votes
1answer
67 views
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 \;\;\; \&...
0
votes
1answer
29 views
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 ...
2
votes
1answer
42 views
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 ...
1
vote
2answers
43 views
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 ...
0
votes
0answers
10 views
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 ...
4
votes
2answers
71 views
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$(X_i, Y_j) = 0$?
6
votes
1answer
262 views
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 ...
2
votes
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?
$o_p(X_n) + o_p(Y_n) = o_p(|X_n - Y_n|)$
0
votes
0answers
10 views
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 ...
3
votes
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 ...
0
votes
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 ...
4
votes
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 ,...
0
votes
0answers
49 views
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 ...
0
votes
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, "...
0
votes
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
...
1
vote
0answers
18 views
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 \...
1
vote
1answer
23 views
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}^...
1
vote
0answers
9 views
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 ...
3
votes
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{...
8
votes
3answers
107 views
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 $...
0
votes
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)^...
1
vote
0answers
51 views
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,...
