Questions tagged [random-variables]
Questions about maps from a probability space to a measure space which are measurable.
11,649
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$\limsup_{n\to\infty}S_n=+\infty\ \mathrm{a.s.}$ and $\liminf_{n\to\infty}S_n=-\infty\ \mathrm{a.s.}$ for the random walk $(S_n)$?
Let $(X_n)_{n\geq1}$ be i.i.d. real-valued random variables such that $\mathbb{E}(X_1)=0$ and $\mathrm{Var}(X_1)=\mathbb{E}({X_1}^2)>0$, where the variance $\mathrm{Var}(X_1)$ can be $+\infty$. Let ...
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Expectation of a function of three random variables
I have three random variables $X, Y,$ and $Z$. $X=a$ whenever $Y \geq b$, and $X=0$ whenever $Y<b$, where $a$ and $b$ are constants. I have some function $g(X,Y,Z)$ of all three variables. I know ...
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Random vectors with the same distribution
I found the following question in my lecture notes on probability theory: suppose for some stochastic processes $\{X_t\}$ and $\{Y_t\}$ $\forall t_1,t_2$ $(X_{t_1},X_{t_2})\overset{d}{=}(Y_{t_1}, Y_{...
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expected value of mixed RV
I am trying to find a formula to find the expected value of a RV that is mixed (continuous and discrete).
I wrote the probability as a convex Combination of a discrete and absolutely continuous ...
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Probability distribution of resampled quantile
There is a list $X$ of $N$ different numbers, sorted in ascending order. First, we perform resampling with replacement, constructing a list $Y$. This means that we consider a uniform distribution over ...
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40
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Equality in Distribution for Suprema of Uncountably Many Random Variables
According to this post, if $(X_t)_{t\in T}$ and $(Y_t)_{t\in T}$ are equal in distribution and if $T$ is countable, then their suprema $\sup_{t\in T}X_t$ and $\sup_{t\in T}Y_t$ are equal in ...
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Can two random variables have linear relationship but still be uncorrelated?
If two random variables does not have linear relationship then they are uncorrelated even if they might be dependent. I am unable to come up with an example where two random variables have linear ...
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Dice throws - show that expected values are equal
We throw a six-sided die independently.
Let be $X_1$ a random variable (rv) that counts all throws until we see the the first time two consecutive $6$'s. Let be $X_2$ the rv that ignores the first ...
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Difference between $P(f(x*,w)>0)→1,P(f(x,w)>0)→1$ and $P(min(f(x*,w),f(x,w))>0)→1$ when dimension grows
Let $f(x_1,\cdots,x_n,w)$ be a function from $R^{n+1}\rightarrow R$, where $x_1,\cdots,x_n$ are deterministic variables, and $w$ be random variable. As a simple example, $f$ can be $(x_1+\cdots+x_n)w$....
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Are two random variable $X, Y$ independent?
Problem :
Let $X$ be a random day of the week, coded so that Monday is 1, Tuesday is 2, etc. (so $X$ takes values 1, 2, . . . , 7, with equal probabilities). Let $Y$ be the next day after $X$ (again ...
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Probability of the minimun value of n geometric variables be y
I already have the answer to this problem but i would like to understando if and why my resolution is correct or no. Could anyone help me, please?
Question: Let $X_1, X_2, X_3, \dots X_n$ be ...
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If $E|Y|\lt \infty$, $E[Y|X] = m(X)$ and $(X_1,Y_1)$ come from same distribution as $(X,Y)$. Is it true that $E[Y_1|X_1] = m(X_1)$?
If $X,Y$ are random variables, such that $\mathbb E|Y|\lt \infty$, $\mathbb E[Y|X] = m(X)$ where $m$ is some Borel measurable function and $(X_1,Y_1)$ come from same distribution as $(X,Y)$. Is it ...
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On the calculation of chi-square statistics for variables following a multinomial distribution
Setup
$X=(X_1, X_2, \cdots, X_k) \sim M_N(n; p_1, \cdots, p_k),\ n = N_1+ \cdots +N_k$ and
$$
\chi^2 := \sum_{i=1}^k \frac{(X_i - np_i)^2}{np_i}
$$
Fact
$$
\frac{\mathbb{E}[(X_i - np_i)^4]}{(np_i)^2} =...
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Variance of Brownian motion increments
One of the conditions for a family of random variables
$W =(W_t, t \geq 0)$ to be Brownian motion is that the increments are normally distributed, specifically $W_t - W_s \sim N(0, t-s)$, for every $0 ...
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Conditional probability when the given outcome has probability $0$
Consider two different random variables on $\{0,1\}^{\mathbb N}$, i.e. the set of binary sequences. The first random variable $X_1$ has a distribution defined by letting each of its digits be chosen ...
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Finding the differential entropy for random variable
I'm having trouble understanding how to approach this problem, and would like to get some help and explanations :)
Let $X$ be a random variable with density $f_X(x)$. It is known that $X$ is non-...
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Linear combination of two Brownian Motions
Let $W_1(t)$ and $W_2(t)$ be two independent Brownian motions.
Define the new process $X(t) =(W_1(t) - W_2(t))/√2$. Is $X(t)$ then another Brownian motion? I.e check that
1.$X(0) = 0$.
2.$X(t)−X(s)$ ...
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Probability of $Y > X$ between random variables
Set up
r.v. $X, Y \sim i.i.d. F$, and $X, Y$ are Discrete random variables.
Question
$$
P(Y > X) \overset{?}{=} \sum_x \mathbb{P}(Y > x) \mathbb{P}(X = x)
$$
On what basis can this formula be ...
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How do you calculate the Expectation $E(X^4)$? [duplicate]
Suppose $X$ is a normal random variable with, $\mu_x\ {\ne}\ 0\ \text{or}\ 1$ and $\sigma_x\ {\ne}\ 0\ \text{or}\ 1$. I would like to show that the expectation $E(X^4)$ has the following equality: $E(...
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Converge in probability
I'm trying to solve the following Probability problem:
Consider $ X(\omega) : [0,1] \to [0,1] $, where $f(x) = 2x \mathbb{I}_{[0,1/2]}(x) + 0.75 \mathbb{I}_{\{1\}}(x) $ and $X(\omega) = \omega = x$. ...
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Total variation distance bound for two Poisson Binomial distributions with similar parameter vectors
Let $X = \sum_{i=1}^n X_i$ and $Y = \sum_{i=1}^n Y_i$ be two Poisson Binomial distributions (PBDs) with respective parameter vectors $p = (p_1, p_2, \dots, p_n)$ and $q = (q_1, q_2, \dots, q_n)$. ...
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Show that $\mathbb{E}\left(|X|^q\right) \leq \left[\mathbb{E}\left(|X|^p\right)\right]^{\frac{q}{p}}$
Let $X$ be a random variable and let $p \in (0, \infty)$ such that $\mathbb{E}\left(|X|^p\right) < \infty$. Show that for all $q \in (0, p)$, we have $\mathbb{E}\left(|X|^q\right) \leq \left[\...
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mathematical notation for expectation through stochastic function of random variable.
suppose I have random variable $\mathcal{D}$ and I want to compute the estimate over this then I write
$\mathbb{E}[\mathcal{D}]$
If i want to compute the estimate of this random variable passing ...
-1
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2
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independence of random variables with random index
Let $\{X_i\}$ be iid real random variables. And N be a random variable, taking values in $\mathbb{N}$.
My question is if $X_{N}$ has the same distribution as $X_{N+1}$, because of the iid assumption? ...
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Conditional independence vs independence
Suppose I have a set of random variables $\left(A_j\right)_{j=1}^n$ such that
$\mathrm{Prob}[A_n=a_n;\ldots; A_1=a_1|A_0=a_0]=\prod_{j=1}^n\mathrm{Prob}[A_j=a_j]$
Are they independent of one another ...
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The distribution of sum of i.i.d random variables from Laplace Distribution
From a Laplace distribution $Laplace(0, b)$, we randomly pick one sample from the distribution and repeat $n$ times. Then, we sum these values and get a sum $s = \Sigma_{i=1}^n x_i$ where $𝑥_𝑖$∼$...
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Multiple coin toss with elimination
$K > 1$ balanced coins are tossed simultaneously and independently. After this we proceed to eliminate all the heads and keep the rest. The remaining coins are tossed again, also simultaneously and ...
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Expected value of a discrete random variable $E[X] = \sum_{x=1}^\infty P(X\geq x)$ [duplicate]
Question
Let $X$ be a random variable that takes positive integer values.
When E[X] exists, I want to show
$$
E[X] = \sum_{x = 1}^\infty P(X \geq x).
$$
What I know
Let $f$ be the probability function ...
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Expectation Conditioned on $\sigma$-subalgebra
In preparing for my upcoming qualifying exam, I have encountered the following problem:
Consider the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ where $\Omega = [0,1]$, $\mathcal{F}$ is the ...
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Computing the sum of two i.i.d uniform random variables inside a unit disk?
I'm trying to find the sum of two independent variables $A$ and $B$ whose densities are defined as follows:
$$f_A(x,y)=\frac{1}{\pi}, \sqrt{x^2+y^2}\le1$$
and
$$f_B(x,y)=\frac{1}{\pi}, \sqrt{x^2+y^2}\...
3
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Modulus of characteristic function equal to 1 implies almost surely constant
Let $X$ be a random variable, and let $\hat{\mu_X}$ be its characteristic function.
Suppose that $|\hat{\mu_X}(u)| = |\hat{\mu_X}(v)| = 1$ for some $u,v \in \mathbb{R}^*$, with $uv^{-1} \not \in \...
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ballistic problem
A projectile is launched at an angle $θ$ with respect to the surface with velocity $v_0$ (deterministic). If the angle of inclination is a uniform random variable on $[0,π/2]$, calculate the function ...
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How to prove finite absolute value of expectation for $Z(t)=W(t)^2t$ when $W(t)$ is a standard Wiener Process? [closed]
The question lies in the title. I am trying to show that $Z(t)$ is a Martingale. The martingale property associated with the filtration I have computed..
I tried using the Hölder's Inequality to prove ...
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Random variable $f(N_\lambda) $ with same law as Poisson $N_\lambda$
Set $S_n(p) $ random variable which follows the binomial distribution of parameters $n$ and $p$, that is
$$P(S_n(p) =k) =\begin{pmatrix} n\\k\end{pmatrix}p^k(1-p)^k. $$
It is clear that $n-S_n(1-p)$ ...
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Uniform random triangles in the square.
Select $3$ points in the unit square $[0,1]^2$ randomly using the uniform distribution.
That gives you a triangle. What is the expected distribution of edge lengths and internal angles?
You could ask ...
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Pdf of the difference between the maximum and minimum of uniform random variables
Let $\{X_{i}\}_{i=1}^{n}$ independent random variables with uniform distribution in the
interval $[0,1]$. Let us denote by $\displaystyle U=\max_{i=1,\ldots,n}X_{i}$ and $\displaystyle V=\min_{i=1,\...
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2
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Intuition and use of CDF when we either have PMF or PDF with us [closed]
When we speak about Discrete random variables we have PMF. When speaking about Continuous random variable we have PDF. So my question is if we already have PDF or PMF with us then why do we use or ...
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Prove that $E(F(X))=\frac{1}{2}+\frac{1}{2}\sum_x[P(X=x)]^2$
Suppose $X$ is a random variable with CDF $F$. Prove that $$E(F(X))=\frac{1}{2}+\frac{1}{2}\sum_x[P(X=x)]^2$$
It was straightforward to prove that $E(F(X))=\frac{1}{2}$ for a continuous random ...
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Show that $E(Y)-E(X)=\int_{-\infty}^\infty[P(X<t\le Y)-P(Y<t\le X)]\lambda(dt)$ [duplicate]
If $X,Y$ are defined on the same probability space and have finite expectations, show that $$E(Y)-E(X)=\int_{-\infty}^\infty[P(X<t\le Y)-P(Y<t\le X)]\lambda(dt).$$
My attempt:
\begin{align}
E(Y)-...
- 1,893
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Show that $\sum_{n=1}^\infty P\{X\ge n\}\le EX\le 1+\sum_{n=1}^\infty P\{X\ge n\}$. [duplicate]
Let $X$ be a non negative random variable. Show that $$\sum_{n=1}^\infty P\{X\ge n\}\le EX\le 1+\sum_{n=1}^\infty P\{X\ge n\}.$$
Now, the thing is I need this to prove the next part of the question, ...
- 1,893
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42
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Expected number of attempts until all values from a set are randomly obtained
Say we have a set of $n$ values. At each attempt, one of these can be obtained randomly, with equal probability ($1/n$). The set doesn't change, so a previously obtained value can be gotten multiple ...
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2
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Is $\mathbb{E}[|X|] \geq |\mathbb{E}[X]|$? [duplicate]
Given a random variable $X$, from the definition of the variance, $\mathbb{E}[X^2] \geq \mathbb{E}[X]^2$. It seems intuitive that $\mathbb{E}[|X|] \geq |\mathbb{E}[X]|$ should be true. Is this the ...
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Distribution of $\epsilon(X+Y) + \gamma(X+Z)$ with $\epsilon, \gamma\sim{\mathcal U}(0,1/2)$
I wish to calculate the distribution for the following combination of random variables
$$\epsilon(X+Y) + \gamma(X+Z)$$
Where $\epsilon$ and $\gamma$ are random variables drawn from an uniform ...
- 121
1
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1
answer
51
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Distance from origin when (X,Y) are uniformly distributed
Let's say that $X$ and $Y$ are two independent random variables with distribution $U(0,c)$. If a point on the Cartesian plane is defined by $(X,Y)$, then the distance from the origin is the random ...
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1
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123
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How to calculate the density function?
Let $(X, Y, Z)$ be a randomly absolutely continuous vector, whose density function is
given by:
$$f_{(X,Y,Z)}(x,y,z)=\frac{6}{(1+x+y+z)^4}$$
For $x,y,z>0$. Calculate the density function of $W=X+Y+...
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1
answer
27
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A general moment made up from a pair of independent Gaussian random variables $A$ and $B$ with identical variance factorizes?
Let $A$ and $B$ be real-valued, centered and independent Gaussian random variables such that
\begin{equation}
Cov(A,A)=Cov(B,B):=C
\end{equation}
Since they are independent, it is clear that
\begin{...
- 6,080
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0
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18
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Expression of sum over a point process
I am given a sample of real random variables of the form $(X_1,A_1,\dots,X_N,A_N)$. Let $N_{x,a}=\sum_{t=1}^{N}\mathbb{1}_{X_t=x,A_t=a}$ and for every measurable set $E \subset \mathbb{R}$, let $\...
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1
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29
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Sub-gaussian norm of linear combination
Let $X_1, X_2, \ldots, X_N$ be independent subgaussian random variables, and let $a_1, a_2, \ldots, a_N$ be nonnegative constants.
Consider the random variable $Y = \sum_{i=1}^N a_i X_i$.
I want to ...
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If $V(X)=0$, show $P\{X=m\}=1$ for a rv $X$ with $EX=m$
Suppose $X$ is a random variable with $EX=m$ and $V(X)=\sigma^2$. If $\sigma^2=0$, show that $P\{X=m\}=1$.
Following the Lebesgue integral definition of $V(X)$ and the property that if the integral of ...
- 1,893
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0
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The expected value and standard deviation of parking fee
In order to calculate the parking fee, mayor of the city is using a random variable $X 〜 Τ([0; 25]).$ The value of $X$ is used to calculate the parking fee $Y = max(X - 7, 0)$. Find the expected value ...
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