Questions tagged [random-variables]
Questions about maps from a probability space to a measure space which are measurable.
8,622
questions
0
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0answers
28 views
Are “independent events” in probability really independent? [closed]
This is a hard and deep question. I understand very well the concept of independence. But, let us take two events: Event A (I throw a dice) and event B (some star explodes in an near galaxy). Are ...
0
votes
0answers
11 views
Find the limit cdf for a sequence of random varibales whose characteristic functions do not converge.
X is a random variable with $P(x=-1)=P(x=1)=\frac{1}{2}$, $Y_n=nX$ is a sequence of random variables, n=1,2, $\dots$ $S_n = \frac{1}{n}\sum_{i=1}^{n} Y_i$ is the partial sum of $Y_n$. I am asked to ...
1
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0answers
42 views
Problem with module
Knowing that…
$Y:=\left\{\begin{matrix} X & if & |X|<a\\ -X & if & |X| \geq a \end{matrix}\right.$;
$X \sim N(0,1)$;
$a>0$;
…and obviously $|X|:=\left\{\begin{matrix}
X & ...
0
votes
1answer
21 views
General form of the open mapping theorem
Let $X,X_1,X_2,...$ be real valued random variables on the same probability space $(\Omega, \mathcal{F},\mathbb{P})$. Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function.We know that ...
0
votes
1answer
18 views
Does the Martingale Representation theorem hold both ways?
Can the Martingale Representation theorem be used to assume that the integral with respect to Brownian motion, $B(t,\omega)$,
$$X=\int^{T}_{0}B^{4}(t,\omega)dB(t,\omega)$$
is a square integrable ...
0
votes
1answer
20 views
Density of $g(Y)=\frac{1}{2}\mathbb{E}[X|Y]$
Let $(X,Y)$ a random variable with density $f(x,y)=cx(y-x)e^{-y}$ for $0 \leq x \leq y <\infty$. Find:
1) the value of $c$.
$\rightarrow c=1$
2) the density of $X|Y=y$.
$\rightarrow f_{X|Y}(x,y)...
2
votes
1answer
38 views
What's the expectation of the function given by $|X_1|$ when $X=(X_1 \dots X_d) \sim Unif(S^{d-1}(r))$
Let $X=(X_1 \dots X_d) \sim Unif(r S^{d-1} \equiv S^{d-1}(r) ).$ I'd like to know what's the expectation of $|X_1|,$ i.e. what's the integral:
$$ \frac{1}{vol(S^{d-1}(r))} \int_{S^{d-1}(r)} |x_1| d ...
-1
votes
0answers
14 views
Show the convergence of $\sum_{i=1}^{n}\prod_{i=1}^{n}(X_{i})$ [closed]
P($X_{i}$=a)=P($X_{i}$=b)=0.5;
a*b=0.5;
$Y_{n}$=$\sum_{i=1}^{n}\prod_{i=1}^{n}(X_{i})$;
Does {$Y_{n}$}$_{n=1}^{∞}$ converge a.s. to some limit random variables? If yes, what is its mean and ...
0
votes
0answers
24 views
What is the variance of $\frac{1}{n-1}\sum_{i=1}^{n-1}(x_{i+1}-x_i)^2$?
Consider a length $n$ vector $\mathbf{x}$ containing $n$ i.i.d. observations $\{x_i\}_{i=1}^n$ of a random variable $X$ with zero mean and unit variance. Let $\mathbf{z}$ be a length $n-1$ vector ...
0
votes
1answer
31 views
Finding the mean sales price
everyone.
I have the following problem: a factory produce valves, with 20% chance of a given valve be broken. The valves are sold in boxes, containing ten valves in each box. If no broken valve is ...
-2
votes
0answers
22 views
The following is my question. Thank you! [closed]
Suppose that a particle is released at the origin of
the xy-plane and travels into the half-plane where x >0.
Suppose that the particle travels in a straight line and that
the angle between the ...
2
votes
1answer
41 views
Probability problem about repeated permutations
I have a box with $n$ red balls and $n$ blue balls, and I pick them one by one. When the color of the ball that I picked is different from the color of the previous ball, I earn a point. Let $X$ be ...
1
vote
0answers
17 views
Negative moments of variables
Let $f_i, i=1, \ldots, n$ be independent Steinhaus random variables, i.e. variables which are uniformly distributed on the complex unit circle. Let $a \in R^n$.
Find $E\left(\sum_{i=1}^nf_i a_i\...
1
vote
1answer
17 views
Dependence of two random variables
In a Bernoulli experiment of parameter $p$ let $T$ be the instant of first success and $U$ the instant of second success. Find the density of $U$ and tell if $T$ and $U$ are independent or not.
This ...
0
votes
1answer
19 views
a uniform random variable and its limit [closed]
I found this exercise on my book but I don't understand how to solve it:
Let $U_n$ be an uniform random variable in $\{1,2,...,n\}$ and let $V_n=\frac {U_n}{n}$.
Show that if, $0 \lt a \lt b \lt 1$ ...
0
votes
0answers
20 views
How to allow a genetic algorithm to build a generic mathematical formula?
Assume I have 4 known numbers, all in a 0-400 range, as follows:
Variable1 Variable2 Variable3 Variable4
0-400 0-400 0-400 0-400
I also ...
0
votes
1answer
26 views
$X \sim U[-1,1]$, is there $Y$ independent of $X$ s.t $X+Y$ and $\frac{Y}{2}$ have the same distribution?
I have thought that if there exist such $Y$ then we can look at characteristic functions of $X + Y$ and $\frac{Y}{2}$ to get:
$$ \phi_{X+Y}(t)= \phi_X(t)\phi_Y(t) = \phi_{\frac{Y}{2}}(t) \\ \frac{\...
0
votes
2answers
22 views
Almost sure convergence of random variables with same mean and the difference goes to zero on the product
Let $X_n$ be a sequence of independent real valued random variables on the same event space, with the same (finite) mean $\mu$.
Suppose that for almost every couple of points $(\omega,\omega')$ in ...
0
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0answers
20 views
Understanding the Monte Carlo Equation which returns value to the expected function
I was reading an article related to Monte Carlo Method. The link of the article is:
Monte Carlo Lecture
I have following questions:
1)In the equation related to the attached image, we are assigning ...
3
votes
3answers
110 views
Understanding a common proof for linearity of expectation
For any two discrete random variables $X,Y:\Omega \to \mathbb{R}$ in a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, linearity of expectation tells us that: $$\mathbb{E}(X+Y) := \sum_{t \in (...
0
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0answers
14 views
Is there a meaningful additive risk measure
An important property of coherent risk measures is subadditivity. But are there any additive risk measures that can be used in a meaningful way? (I would exclude the expectation for example)
0
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0answers
6 views
Continuous variable function that evaluates to random curves
There exists a function of a continuous parameter function that sometimes evaluates to one curve and other to a different one? For instance, is there a function $f(x)$ that say is 0 for $x < 0$ and ...
-1
votes
1answer
19 views
Two independent random variables X and Y are given. How can we compute Expectation minimum {X,Y} and Expectation maximum {X,Y}? [closed]
Two independent random variables X and Y are given. X is uniformly distributed in the interval [-2,5], Y is uniformly distributed in the interval [-4,2]. How can we compute Expectation minimum {X,Y} ...
0
votes
0answers
27 views
Let $m ≥ 1$, then I have to find $C_m > 0$ such that $f(n)$ = $\frac{C_m}{n}$ $−$ $\frac{C_m}{n+1}$ is a pmf on {${m,m+1,…}$}
I really could use some help with the following questions regarding Probability Theory. I really do not know where to start.
Let $m ≥ 1$, then I have to find $C_m > 0$ such that $f(n)$ = $\frac{...
0
votes
0answers
33 views
Formalising some very basic probabilistic problem
Often, I encounter some problem were probability is involved. On an intuitive level, I understand I think what is going on but when I have to formalize it with probability spaces I am a bit lost.
For ...
0
votes
1answer
29 views
Does the convergence in probability imply the following limit is $1?$
Let $X_n \in \mathbb{R}$ be a sequence of non-constant random variable with continuous PDF converging in probability to $c,$ but not necessarily convergence almost surely, i.e.
$$\lim\limits_{n \to \...
0
votes
3answers
58 views
Distribution of $\Big(Y_1+Y_2\Big)^2$ and $\Big(Y_1-Y_2\Big)^2$ where $Y_i \sim N(0,1)$
Does anyone know what is the distribution of $(Y_1+Y_2)^2$ and $(Y_1-Y_2)^2$ where $Y_i \sim N(0,1)$ are independent variables? I have tried to go through the joint pdf, but when trying to change ...
0
votes
0answers
42 views
+100
An upper bound for $\sum_{n=1}^{\infty} n^{r-2} P\{|\sum_{k=1}^{n} \sum_{i=-\infty}^{\infty} a_{i} X_{i+k}|>\varepsilon n\}$
Let $1 \leq r<2$ and let $\left\{X, X_{i},-\infty<i<\infty\right\}$ be a sequence of pairwise i.i.d. random variables. Let $\left\{a_{i},-\infty<i<\infty\right\}$ be a sequence of real ...
0
votes
2answers
25 views
Brownian motion increments - are they random variables or random processes
If $W_t$ is a Brownian motion process and $0 \le t_1 \le t_2$ then is the increment $W_{t2} - W_{t1}$ a random variable or a random process? My lectures say "random variable" but I believe it makes ...
1
vote
1answer
29 views
Almost sure convergence of a sum of independent r.v
Let $S_n:=\sum_{i=1}^nX_i$ where $X_1,X_2,...$ are indepentent r.v.'s
such that:
$P(X_n=n^2-1)=\frac{1}{n^2}$ and $P(X_n=-1)=1-\frac{1}{n^2}$
Show that $\frac{S_n}{n}\rightarrow-1$ almost ...
0
votes
1answer
50 views
How to solve the below limit by means of L'H$\hat{\text{o}}$pital rule?
Let $(X_n)_{n\geq1}$ be a sequence of random variables satisfying:
i) $\mathbb{E}\{X_n|\mathcal{F}_{n-1}\}=0$;
ii) $\mathbb{E}\{X_n^2|\mathcal{F}_{n-1}\}=1$;
iii) $\mathbb{E}\{|X_n|^3|\mathcal{F}_{n-1}...
1
vote
1answer
29 views
Inequality for the expectation of a nonnegative random variable
Let $X$ be a nonnegative random variable with distribution $F$ and mean $\mu=E(X)>0$. Let $A_{\mu}=[\mu, \infty)$. Is it true that
$$
\int_{A_\mu} x dF(x) \geq \mu/2
$$
must then hold?
I'm trying ...
1
vote
0answers
62 views
Probability to hit a set of pairs
Let $[n] = \{1, \ldots, n\}$, and suppose that $X = \{(i,j) : i< j\}$ is some subset of $[n]^2$, and we are given only its size $|X| = x$.
Now assume I sample $y$ elements uniformly at random ...
0
votes
1answer
37 views
Is this identity on derivatives of random variables valid?
Is this identity valid? $a$, $b$, $c$ are all random variables.
$$da = \int(\int d(a|b,c) db)dc$$
1
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0answers
19 views
Sigma-algebra generated by a sequence with zeros
Let $\{x_i\}_{i\in\mathbb{N}}$ be a sequence of zero mean random variables on a probability space $(\Omega, F, P)$. Suppose that there is some $i_0>0$ such that $x_i=0$ for all $i\geq i_0$.
Does ...
0
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0answers
19 views
Show that : $\sum_{n=1}^{\infty} n^{-3} \sum_{k=1}^{n} E|X_{k}|^{2} I(|X_{k}| \leq n) \leq c \sum_{k=1}^{n} E|X_{k}|^{p} \sum_{n=k}^{\infty} n^{-1-p}$
I was trying to add some details to a result concerning the strong law of large numbers for pairwise independent random variables : here's what I've done :
Let $\epsilon > 0,1<p<2, \, (X_i)_{...
0
votes
1answer
15 views
How do I obtain the pdf of a random variable, which is a function of random variable.
A random variable, $X$, has a value of zero with probability $1/3$, and follows a uniform distribution over $[-1, 1]$ with probability $2/3$. How can I derive the pdf of $X$?
In my opinion, $X$ can ...
0
votes
1answer
36 views
Is it true that $\mathbb{E}\{X\}=\mathbb{E}\{Y\}$ $\Rightarrow$ $\mathbb{E}\{X|\mathcal{F}\}=\mathbb{E}\{Y|\mathcal{F}\}$?
Given $(\Omega$, $\mathcal{F}$, $\mathbb{P})$ and two r.v.'s $X$ and $Y$ defined on it, does it hold true that:
$$\mathbb{E}\{X\}=\mathbb{E}\{Y\}\Rightarrow\mathbb{E}\{X|\mathcal{F}\}=\mathbb{E}\{Y|\...
0
votes
0answers
19 views
What is the CDF of a linear combination of a Gamma and a Hyperbolic random variables?
I am interested in the evaluation of the risk stemming from the combination of two weather events: heavy rain and low temperature. Rainfall $R$ can be modeled as a Gamma distribution, temperature $T$ ...
0
votes
2answers
24 views
Proof that two iid Gaussian random variables are conditionally independent of their mean given their sum
My question is a simplification of a statement in this book that i.i.d. Gaussian random variables $X_1, X_2, ..., X_n \sim \mathcal{N}(\Theta, 1)$ are conditionally independent of $\Theta$ given their ...
0
votes
3answers
37 views
Doubt about random variables and sample spaces
I have this little problem, reviewing my notes from the university of the first days of probability, I realized that there was a blank problem in my notes, says the following:
"
In a bag there are 5 ...
0
votes
3answers
50 views
If $X$ is a random variable, and $Y= 2X$, then why isn't it enough to multiply the density function of $X$ by $2$ to find the density function of $Y$?
This may be a dumb question, and I've tried searching online for answers, but I can't seem to wrap my head around it.
So say I have a random variable $X$ and $Y = 2X$. Now I want to find the density ...
1
vote
1answer
23 views
No sequence of rv's such that $X_n\overset{P}{\rightarrow}0$ and $\mathbb{E}(X_n)\to 2$ and also $\sup\mathbb{E}(X_n^2)<\infty$.
Prove that there is no sequence $(X_n)$ such that $X_n\overset{P}{\rightarrow}0$ and
$\mathbb{E}(X_n)\to 2$ and also $\sup_n\mathbb{E}(X_n^2)<\infty.$
Attempt. If we didn't have the last ...
0
votes
0answers
15 views
$\min(S^3,T)$ with $S \perp T$
Let $S$ and $T$ two random variables with exponential distribution of rate $\lambda$ and density $f(u)=\lambda e^{-\lambda u},u>0$. Find the density of:
1) $X=|S-T|$.
$\rightarrow X\sim Exp(\...
0
votes
0answers
9 views
Stopping time for sum of draws from a uniform random variable [duplicate]
If you generate $u1,u2..$ from a uniform distribution in $[0,1]$. What is the average stopping time $T$ for the sum $(u1+..+uT)$ to be greater than $1$?
Apparently the sum of random variables forms ...
2
votes
1answer
45 views
$\mathbb{P}(X>Y)$ for $X,Y$ two Poisson
Two teams have to play a final of a tournament. Team A score a number of goals that can be shaped like a random variable $X \sim Poi(\lambda_{A}=2.5)$. Team B score a number of goals that can be ...
0
votes
0answers
3 views
Independence relation for joint filtrations
Let $X$ be a random variable with first moment.
Let $\mathcal A$ and $\mathcal B$ be sub-$\sigma$-algebras.
Let $\mathcal A$ be independent of $\sigma(X) \vee \mathcal B$.
Does it hold that?
$$
E( X ...
1
vote
0answers
31 views
In what sense are independent random vectors “independent”? [duplicate]
Let's say we have independent random vectors $\boldsymbol{X}$ and $\boldsymbol{Y}$, where $\boldsymbol{X}=(X_{1},...,X_{p})$ and $\boldsymbol{Y}=(Y_{1},...,Y_{q})$. What is it that makes them ...
0
votes
1answer
39 views
Entropy of the Uniform Mixture of Discrete Probability Distribtuions
Consider the following inequality:
\begin{equation}
H\left(\frac{1}{3}p_{1} + \frac{1}{3}p_{2} + \frac{1}{3}p_{3}\right) \geq H(0.5p_{1} + 0.5p_{2})
\end{equation}
where H(.) denotes the Shannon ...
0
votes
0answers
28 views
Expectation of maximum over an infinite sequence of sub-Guassian random variables
It's a problem from 'High Dimensional Statistics' (MIT's lecture notes)
(Problem 1.3,page30)Let $X_1,X_2$ ... be an infinite sequence of sub-Guassian random variables with variance proxy $\sigma_{i}^...
