Questions tagged [random-variables]
Questions about maps from a probability space to a measure space which are measurable.
6,182 questions
0
votes
0answers
26 views
Probability theory - multivariate random variable(computational problem)
Question: Given the following covariance matrix of symmetric gaussian multivariate random variable $(X,Y)$ to be of the form:
\begin{bmatrix}
2 & 1 \\
1 & 1
\end{bmatrix}
...
-2
votes
1answer
38 views
Conditional Expectation of $\mathbb E[X^4 | X]$ [on hold]
We know that $\mathbb E[X|X] = X$ but what about $\mathbb E[X^4 | X]$?
1
vote
0answers
31 views
Uniform Distribution Probability of 3 Movie times
The question is as follows:
A short film will show three times tomorrow - at $12:00, 13:30$ and $15:00$. If you arrive at the theatre at a random time between $12:30$ and $14:30$, what is the ...
0
votes
0answers
33 views
Maximum of two random variables
Say we have two random variables $X,Y$ (that is all we know) and a new random variable $Z = \max(X,Y)$
Those random variables have distribution functions $F^X$, $F^Y$ and $F^Z$.
The question where I'...
0
votes
2answers
14 views
Convolution of weighted uniformly distributed random variables
I have problems with deriving the CDF of a weighted difference of iid uniformly distributed random variables.
Assume that $X_1$ and $X_2$ $\stackrel{iid}{\sim} U[0,1]$. Define Z = $a\cdot X_1 - X_2$, ...
0
votes
1answer
28 views
Notation for random function?
Say I have a sample space $\Omega$. I want for every $\omega \in \Omega$, a function $f: A\to B$, for some arbitrary sets $A,B$. (note this is not in the context of random walks, or sample statistcs ...
0
votes
1answer
47 views
$L^2$ Bound for Random Variables
Let $X$ be an $L^2$ random variable. Show we can establish the following bound:
$$
\mathbb{P}(|X| \geq \gamma \mathbb{E}[|X|]) \geq (1- \gamma)^2\frac{\mathbb
E[|X|]^2}{\mathbb{E}[X^2]}
$$
for $0&...
3
votes
1answer
45 views
Independent sums of independent random variables
Suppose $X=X_1 + X_2$ and $Y=Y_1 + Y_2$ are independent random variables such that $X_1,X_2$ are independent and $Y_1,Y_2$ are independent. Does this imply that $X_i,Y_j$ (for $i,j\in \{1,2\}$) are ...
0
votes
1answer
16 views
Variance of squared $l_2$ distance ratio
Problem: Assume we have an i.i.d data set $\{\bar{x} \} \subset \mathbb{R}^n$, sampled from an $n-$dimensional normal distribution where each dimension is independent, with zero mean and $\sigma^2$ ...
2
votes
2answers
37 views
The expected value of $C$ is equal to $\frac{a}{b}$ for coprime positive integers $a$ and $b.$ What is $a+b?$
A fair, $6$-sided die is rolled $20$ times, and the sequence of the rolls is recorded.
$C$ is the number of times in the 20-number sequence that a subsequence (of any length from one to six) of rolls ...
0
votes
0answers
12 views
What is a permutation-valued random variable
I am reading in wikipedia about random permutation and here appear the next term "permutation-valued random variable". Could you give me a book where exist this definition please? Do you have any ...
-1
votes
0answers
14 views
Help with monotone convergence theorem (RV)
This little problem may be simple thing to an expert in the field but it drives me crazy.
For a sequence of R.V., $0\le X_1\le X_2 \ldots \le X_n \ldots$, I have $\lim_{\rightarrow \infty} E[X_n] = ...
0
votes
1answer
38 views
$E(X)= \int_{x \in \mathbb{R}}{x \cdot f_X(x) dx}$ implies $Eg(X)= \int_{x \in \mathbb{R}}{g(x) \cdot f_X(x) dx}$ on continuous random variables
Let $X$ be a random variable with values in $\mathbb{R}$ and let $g: \mathbb{R} \to \mathbb{R}$ a measurable function such that $ E g(X) $ exist.
If we define the expected value as $E(X)= \int_{x \in ...
0
votes
0answers
28 views
How to take care of expectation in this term?
I need to calculate the result of the following vector, but the problem there is an expectation, and I do not know how to deal with it:
$$ \boldsymbol{\gamma}_k = \text{diag} \left\lbrace \mathbb{E} \...
0
votes
1answer
25 views
Truncated variance less than variance?
Let the random variable X be distributed (not necessarily normally) with some mean $\mu$ and some variance $\sigma^2$. Under what conditions is it true that
$$ var(X|X<a) \leq var(X)$$ for some $a$...
3
votes
0answers
44 views
Show that $Y_1$ and $Y_2$ are independent $N(0, 1)$-distributed random variables.
The random variables $X_1$ and $X_2$ are independent and $\mathcal N(0, 1)$-distributed.
Set
$$Y_1=\frac{X_1^2-X_2^2}{\sqrt{X_1^2+X_2^2}} $$ and
$$Y_2=\frac{2X_1\cdot X_2}{\sqrt{X_1^2+X_2^...
2
votes
1answer
30 views
a.s. convergence uniform distribution
I am having troubles with proving almost surely convergence for the following problem:
Let $U_j$ be IID $U(0,1)$ distributed and define $A_n$ to be:
$A_n=\sum_{k=1}^n \prod_{j=1}^k U_j$ for $n\in \...
0
votes
0answers
18 views
Convergence results for conditional random random variables
Suppose that $X_n$ is a sequence of random variables such that $X_n \rightarrow X$ either almost surely or in $L^p$ or both.
Now consider the sequence of conditional random variables $f(X_n) | X_n \...
-3
votes
2answers
30 views
What is the probability of four people standing back to back in line at a restaurant having the same, common, first name? [on hold]
I'm trying to calculate the odds of this happening because myself and three other men named Steve ended up in line together at a Taco Bell. It seems very strange but maybe it's not as strange as I ...
0
votes
0answers
18 views
Can we come up with a reasonable “expected chance frequency” in this case?
The World Cup winners from 1962 to 2002 read the same going forward as backward, when one allows the 'event' to be either a particular country or a 'host' nation. (Winners Argentina and W Germany were ...
2
votes
1answer
42 views
Trouble with conversion of probability density function
Let X be a uniformly distributed random variable on the interval $(−1, 1)$.
Find the pdf of $Y = X^2$.
So far I know the following:
$$Y\le y = X^2 \le y = X \le y^{1\over2}$$
I also know that $...
-2
votes
1answer
25 views
Random numbers uniformly picked from a range [on hold]
If $x$ is picked randomly uniformly from $\{1,...,N\}$, what can be said about $y=z-x$, if $z$ is any number from $\{1,...,N\}$ ? Will probability of both $x$ and $y$ be same?
0
votes
1answer
52 views
Probability density transformation question
Let X be a random variable that is uniformly distributed in (0,1). Find the
probability density function of Y = −ln X.
I got the solution $e^{-2y}$ for $y>0$, however the real solution is $e^{-y}...
1
vote
1answer
29 views
Number of expected winners in a game involving three kinds of marbles
In a bag there are some red marbles, some blue marbles, and some green marbles.
There is at least one marble of each color in the bag.
Richard, Bob and George ignore the exact numbers of marbles of ...
1
vote
2answers
24 views
Martingale-like properties of extreme value process?
Imagine I have a sequence of i.i.d. random variables $x_1,x_2,...,x_n,...$ and let $M_n = \max(x_1,...,x_n)$. Is there something I can say about the expectation $\mathbb{E}[M_n | M_{n-1}]$? What if $...
3
votes
1answer
27 views
Expected least distance between closest two points out of $n$ drawn from a distribution
Suppose I draw $n$ points in $\mathbb{R}$ from a distribution $p$. What is the expected least distance between two of the points drawn? I am particularly interested in the uniform distribution $\...
1
vote
2answers
30 views
Find pdf of a random variable $Z$ which is a function of two random variables $X$ and $Y$.
Given two arbitrary positive constants $A$ and $B$, and two independent random variables $X$ and $Y$, I want to find out the pdf of $$Z=\frac{A+X}{B+Y}.$$
My process is as follows:
\begin{align}
F_Z(...
1
vote
0answers
17 views
The CDF of the maximum of some function of the maximum two order statistics
Let the random variables $X_1,\,X_2,\,\ldots,\,X_K$ be i.i.d. exponential random variables with parameter 1. Also, let the random variables $Y_1,\,Y_2,\,\ldots,\,Y_K$ be defined similarly.
Now let
$...
1
vote
0answers
39 views
pdf of a function of a normal random variable
Let $f$ be a function $f : x \mapsto y$ where $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^m$; $m \geq n$. $f$ is not invertible. I have a random variable $X$ s.t. $X \sim \mathcal{N}(\mu, \Sigma)$ ...
0
votes
1answer
30 views
Polynomial function of random variable
let $x$ be a random variable with pdf $p(x)$, e.g., $ prob(A)=\int_A p(x)dx$.
Define a random variable $y=f(x)$.
If $f$ and $p$ are polynomial functions, what we can tell about pdf of random variable $...
0
votes
0answers
81 views
+100
Can we split a random variable into intervals on its domain of possible values and express it in terms of “simple” distributions on those intervals?
So suppose we have a random variable $Z$ which can take values in $\left[-A, A \right]$. Suppose we do not know the exact distribution of $Z$.
Now if we take $N$ fixed disjoint intervals of $[-A,A]$ ...
0
votes
0answers
24 views
Sampling from matrix valued distributions
How does random sampling from matrix valued distribution work in general?
In the univariate case we can draw from the uniform distribution over the interval from 0 to 1, then plug that value into ...
2
votes
3answers
96 views
Variance of dot product of two normalized random vector
Given a set of normalized vectors $\mathbf{x}$ and $\mathbf{y}$ of length $N$,
with each entry independently sampled from $\mathcal{N}(0,1)$ before being divided by the vector norm.
By running ...
2
votes
1answer
87 views
Sufficient conditions for continuous functions of continuous random variables to themselves be continuous random variables
I've been trying to figure out nontrivial conditions for continuous functions of continuous random variables to themselves be continuous random variables without much success. Here's what I know so ...
0
votes
2answers
71 views
Expected value for 2 dice roll
Imagine that we roll two fair six-sided dice (i.e., all six sides have equal probability). Let X1 and
X2 be the random variables representing these outcomes.
Now, imagine we take one of the dice rolls,...
2
votes
2answers
174 views
How to find the median of a PDF with a continuous random variable given the mode of it?
So the question is to find the median of $X$ if the mode of the distribution is at $x = \sqrt{2}/4$. And the random variable $X$ has the density function
$$f(x) = \left\{ \begin{array}{ll}
...
-1
votes
1answer
22 views
Stochastic orders of summands when sum has fixed distribution
Suppose $X,X',Y,Y'$ are independent random variables. We know that $X+Y \overset{d}{=} X'+Y'\overset{d}=Uniform[0,1]$ and $X\prec X'$ in the sense that $P(X> x)\le P(X'>x)$ for any $x\in \mathbb{...
0
votes
0answers
13 views
Pointwise $o_p(1)$ and Uniform $o_p(1)$
For every $k=1,...,K$, we have a random sequence denoted by $X_{k}=\{X_{k1},X_{k2},...,X_{kN}\}$ and $X_k=o_p(1)$. Specifically, $\forall k$, $\forall\epsilon>0$, we have that $Pr(|X_{kN}|>\...
1
vote
0answers
26 views
approximating the distribution of sum of products of Binomial random variables
I wonder if there is any way that can allow me to approximate the distribution of the sum of products of Binomial random variables with a closed form?
For a binomial random variable $X_i^{(k)} \sim ...
0
votes
1answer
45 views
Plotting pdf based on value
I'm given a continuous random variable X and have the probability density function as below:
Now, i want to plot when b = 2, and since i wasn't provided with any dataset, i attempted something like:
...
0
votes
1answer
37 views
Converting equations of random variables to distributions
According to this paper on factor analysis, a $p$-dimensional random vector $\textbf{x}$ can be modeled using a $k$-dimensional vector of factors $\textbf{z}$ where $k \ll p$ using this generative ...
3
votes
3answers
96 views
Covariance of polynomials of random normal variables
$\newcommand{\Cov}{\operatorname{Cov}}$If $X$ and $Y$ are random variables with a bivariate normal distribution and:
$X\sim\mathcal{N}(\mu_X,\sigma_X^2)$
$Y\sim\mathcal{N}(\mu_Y,\sigma_Y^2)$
$\Cov(X,...
1
vote
2answers
34 views
Compute $m_Z (t)$. Verify that $m'_Z (0)$ = $E(Z)$ and $m''_Z(0) = E(Z^2)$
Let $Z$ be a discrete random variable with $P(Z = z)$ = $1/2^z$ for $z = 1, 2, 3,...$
(b) Compute $m_Z (t)$. Verify that $m'_Z (0)$ = $E(Z)$ and $m''_Z(0) = E(Z^2)$
$E(Z) = \sum_{z=1}^{\infty} z P(Z ...
0
votes
3answers
58 views
$X\sim \text{Exp}(\lambda)$ use the moment generating function ($m_X(t)$) to find $E(X)$ and $E(X^2)$
Q1) Let $X\sim\text{Exp}(\lambda)$. Find $m_X(t)$.
My attempt:
$$m_X(t) = E[\text{e}^{tX}] = \int_{0}^{\infty}\, \text{e}^{tx} \lambda e^{-\lambda x}\,\text{d}x = \int_{0}^{\infty}\,e^{-\lambda x + ...
-1
votes
1answer
19 views
Expected value of multiplication of two indicator random variables?
I found the following statement in a book:
When $k \neq j$ , the variables $X_{ij}$ and $X_{ik}$ are independent, hence
$E[X_{ij}X_{ik}] = E[X_{ij}]E[X_{ik}]$
where E is the expected value. Can ...
1
vote
1answer
29 views
How does the following equation evaluate?
$$E[n_i^2] = E[(\sum_{j=1}^n X_{ij})^2]$$
$$= E[\sum_{j=1}^n\sum_{k=1}^n X_{ij} X_{ik}]$$
Here $E$ is expected value and $n_i$ is a random variable. $X_{ij}$ is an indicator random variable. How ...
0
votes
2answers
31 views
How to solve the probability of the binomial distribution sequence below?
Let $x_0$, $x_1$...$x_n$ be a sequence of independent random variables. $x_i = 1$ has probability $p$ and $x_i = 0$ has probability $1-p$. Let $k$ be the smallest integer such that $x_k = x_{k+1}$. ...
0
votes
1answer
37 views
What is wrong with my simulation of Laplace` s studying George-Louis Leclerc `s probability of randomly dropped needle of crossing tile lines?
In 1777 George-Louis Leclerc, Comte de Buffon says:
If we drop a needle onto a lined piece of paper, how likely is it to cross one of the lines? If the needle is shorter than the gap between the ...
1
vote
1answer
52 views
Probability of random variables
Imagine that we roll two fair six-sided dice (i.e., all six sides have equal probability). Let X1 and X2 be the random variables representing these outcomes.
Now, imagine we take one of the dice rolls,...
-1
votes
0answers
42 views
Probability density function understanding
Suppose that $X$ is a continuous random variable whose probability density function is given by,
$$ f(x) =
\begin{cases}
4x-2x^2, & \text{ if $0 < x < 2$} \\
0, & \text{ otherwise ...
