Questions tagged [random-variables]
Questions about maps from a probability space to a measure space which are measurable.
9,911
questions
0
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10 views
Characteristic function of gaussian $X + X^2$
Assume $X$ is a standard normal random variable. Are there any known results for the form of the characteristic function of $X+X^2 = X(X+1)$?
Essentially I'm interested in the c.f. of $a + bX + c X^2$ ...
0
votes
1answer
7 views
Describe the following spaces
I'm trying to practice my understanding of $L^1$ and $L^2$ spaces with the problem below.
Given $\Omega = \{\omega_1,...,\omega_7 \} $ let's consider the probability space $(\Omega, \mathcal{F},P)$ ...
-2
votes
0answers
21 views
With random probability
I have a question about the possibility of talking about this problem :\
Let $X,Y$ two random variables, then :
X = a with probability Y and
X = b with probability 1-Y.
Thanks
0
votes
0answers
6 views
Does a non-stationary source has an entropy rate $H_X$?
Let $ \{X_k\}_{k\in \mathbb{N}_+} $ be a source and we assume that $H(X_k )<∞$ for all $k$. Then we define the entropy rate of an information source, which gives the average entropy per letter of ...
0
votes
1answer
23 views
$45$ percentage of population of a large community is in favor of a proposed rise in school taxes.
$45$ percentage of population of a large community is in favor of a proposed rise in school taxes.
$i)$ Approximate the probability that a random sample of $1000$ people will contain at least $500$ ...
1
vote
1answer
20 views
Filling station is being supplied with drinking water once a week. If its weekly volume of sales in thousands of gallons is R.V with $pdf$ given by
PROBLEM: A filling station is being supplied with drinking water once a week. if its weekly volume of sales in thousands of gallons is random variable with pdf given by:
$f(x)= \begin{cases}
5(1-x)^{4}...
0
votes
2answers
27 views
How to add Random Variables in general
I'm kind of confused how to add two arbitrary random variables. For example, suppose I have a binomial random variable $X$ with parameters $(n, p)$ and I have a Poisson Random Variable $Y$ with ...
0
votes
0answers
17 views
Show that $Var(Z|X) < Var(Z|Y)$
Let the random variables $X$, $Y$, $Z$ be $X_t= \epsilon^x_t$, $Y_t= X_t + \epsilon^y_t$ and $Z_t= X_t + \epsilon^z_t$ respectively, and for $E(\epsilon^x)$, $E(\epsilon^y)$, $E(\epsilon^z) \ne 0$.
I ...
0
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0answers
35 views
PDF of sum of (1/(1+uniform distribution)) and a normal distribution
I have a continuous uniform random variable $P\sim U(0,1)$ and a normal random variable $X \sim N(0,\sigma)$. If $Z$ is given by:
\begin{equation}
Z = \frac{1}{1+P}+X
\end{equation}
where $X$ and $P$ ...
0
votes
0answers
8 views
Bounded variance for Lipschitz function of random variable
In Priors for Infinite Networks (Neal, 1996), part of the proof is that $\tanh(X)$ for Gaussian RV $X$ has finite variance, which is later used for the Central Limit Theorem.
For arbitrary activation ...
0
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0answers
17 views
Bounding the CDF of the sum of dependent random variables
Let suppose I have two conitnuous random variables $X$ and $Y$, which are dependent, and I want to find their joint CDF $P(X+Y \leq \alpha)$. Lets suppose I know their marginals $P(X)$ and $P(Y)$, but ...
0
votes
1answer
31 views
In a raffle, $9$ tickets numbered $1$ to $9$ are sold. Two numbers are chosen at random. You hold tickets $1$ to $2$.
In a raffle, $9$ tickets numbered $1$ to $9$ are sold. Two numbers are chosen at random. You hold tickets $1$ to $2$. Find the probability that you:
$a)$win at least $1$ prize
$b)$exactly $1$ prize
...
2
votes
0answers
38 views
Does averaging of random variables always improve concentration?
Let $X_1^{(n)}, X_2^{(n)}, ..., X_J^{(n)}$ be independent and identically distributed random variables satisfying
\begin{align}
\mathbb{E} \left [ X_i^{(n)} \right ] &\leq a_n \tag1\\
\mathbb{P} \...
0
votes
0answers
7 views
Fourier transform of correlated random bits
Suppose I have a sequence of only 1's and 0's randomly chosen, e.g.
1,0,0,0,1,0,1...etc.
If this is a totally random sequence the Fourier transform is a white noise spectrum. But now imagine there is ...
-3
votes
0answers
34 views
How to calculate $f_x$ and the expected value? (Joint probability distribution)
Let x and y be continuous random variables.
$$f_{x,y}(x,y)=\frac{1}{2\pi\sigma_1\sigma_2}\times exp \Bigg\{\frac{-1}{2}\left[(\frac{x-\mu_1}{\sigma_1})^2+(\frac{y-\mu_2}{\sigma_2})^2-2(\frac{x-\mu_1}{\...
1
vote
1answer
33 views
A pair of dice is thrown 180 times in a row. Define random variable $X$
A pair of dice is thrown 180 times in a row. Find the probability that (the total on the two dice is seven), happens at least 25 times.
My question is how can I define random variable $X$ in verbal ...
-1
votes
1answer
34 views
Prove that $P[ X - E [X] \geq a ] \leq P [ | X - E [X] | \geq a]$
I would like to prove that for any RV $X$ and positive value $a > 0$, it holds that:
$$P[ X - E [X] \geq a ] \leq P [ | X - E [X] | \geq a]$$
Intuitively, I understand that $X - E[X]$ can be ...
0
votes
3answers
37 views
Expected value and variance of a set of random variables
Suppose $X_1, X_2, \ldots , X_n$ are $n$ independent r.v.s, with the same probability distribution and with mean $\mu$ and variance $\sigma^2$.
Let $$
\bar{X}=\frac{X_1+X_2+\cdots+X_n}{n}
$$
I know ...
1
vote
1answer
52 views
Does $\mathbb{P}(X_1 > a) \leq \delta$ imply $\mathbb{P}\left (\frac{1}{J} \sum_{i=1}^J X_i > a \right ) \leq \delta$?
Let $X_1, X_2, ..., X_J$ be identically distributed random variables.
Edit: may assume independence as well as finite expectation of the $X_i$'s above.
Does $\mathbb{P}(X_1 > a) \leq \delta$ imply $...
1
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0answers
62 views
Expected value of invariant distribution associated to a random Markov matrix
Let $\Pi$ a random Markov matrix with dimension $m \times m$, that is, its elements are $\Pi_{ij} \in (0,1)$ and its rows sum to one. The rows are independent random variables, i.e. we can write
$$\Pi ...
1
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0answers
18 views
Expressing probability function using two other probability functions (with conditionals).
Let $(X,Y)$ be a discrete random vector, and $X,Y$ are independent random variables, where there exists two non-negative functions, $q,r : \mathbb{R} \to [0,\infty)$ and $a>0$ such that $P_X(x)=aq(...
0
votes
1answer
15 views
Is there a general method for generating correlated random variables from independent random variables?
For generating correlated normal random variables from independent normals, I know that you can use Cholesky/SVD.
Is there a general method that applies for other random variables, e.g., uniformly ...
0
votes
0answers
14 views
Range of correlation between two random variables
Suppose $X, Y, Z$ have mean zero and variance $1$. The correlation between each pair of random variables is $\rho$. What is the range of $\rho$.
I have solve this problem in several ways:
(1)
$$
E[(X^...
0
votes
0answers
9 views
Sampling from a joint distribution with known marginals and dependency structure
Assume that $(X_1, X_2, \dots, X_m) $ are $m$ real-valued random variables with comonotonic dependence. This means by definition that there exists distribution functions $F_1, \dots, F_m$ with ...
0
votes
1answer
27 views
Understanding a convergence proof
I've got a proof for the following result : if $X_n \to X$ in probability then $X_n \to X$ in distribution. The proof goes like this :
Let $t \in \mathbb{R}$ and $\varepsilon >0$. We have $ 1 : \{ ...
0
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0answers
13 views
Known transformations of random variables that lead to a classical distribution?
I was wondering if there's a resource with a consolidated list of transformations, such that when applied to random variables, results in another random variable that is distributed according to some ...
0
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0answers
29 views
Weird question on possibility to create such a Probability Space
I stumbled upon this exercise in my Probability Theory professor's notes:
Determine whether you can create a probability space $(\Omega, \mathcal{F}, P)$ with an infinite amount of random variables $...
1
vote
2answers
23 views
Calculate the marginal [closed]
I want to calculate the marginal $f_X(x)$ of $f_{X, Y}(x, y) = 2 e^{-(x+y)} \mathbb 1_A (x,y)$ where $A = \{(x, y): 0 \leq y \leq x \}$.
I know that I have to integrate $f_{X, Y}$, but how do I handle ...
0
votes
1answer
33 views
During $200$ test demands zero failures were discovered. What is the posterior probability that the software meets the business requirement?
A business software system is required to achieve at most $1$ failure in $1000$ demands (this is equivalent to a probability on demand: $pfd=10^{-3}$). The business manager wants to to evaluate the ...
1
vote
1answer
19 views
Sampling as adding random variables, especially binomial RVs
Is sampling equivalent to adding random variables? I'm a bit confused because as we can see that the binomial distribution becomes more and more shaped like a normal distribution as $n$ increases. We'...
1
vote
1answer
24 views
A producer makes Parts with dimensions between $22$ and $28$. Let $X$ be random variable representing dimension of parts then $X$~$N(25,1.5)$.
A consumer needs parts with specification $26$ $+/-2$ i.e: parts shorter than $24$ and and longer than $28$ are not acceptable. The producer makes parts with dimension normally distributed with mean $...
1
vote
1answer
21 views
Given that the weight of tablets is uniformly distributed between $22$ and $24. $Find the probability that average exceeds $23.5$.
Pharma Tablets are produced in large batches. Individual tablets have weight that is uniformly distributed between $22$ and $24$. A sample of $100$ tablets is randomly selected and the average is ...
1
vote
1answer
43 views
Given the following joint density function; find the expectation of $h(x,y)=2x+5y$
Let $f(x,y) = c(2x+y)$ ; $0<x<2$; and $0<y<3$ and $0$ otherwise
Calculate:
$(i)$ Value of $c$
$(ii)$ Obtain Marginal PDF's of both $X$ and $Y$
$(iii)$ find the expectation of $h(x,y)=2x+5y$...
-1
votes
1answer
48 views
Find E(X) & E(X^2) given that the probability mass function of the variable X is given as follows:
This is my question:
This is my attempt:
Hi, I have tried to solve this problem but I cannot figure it out. Normally, when given a range for x in a function, it would be the Probability Density ...
-2
votes
0answers
23 views
help me resolve this probability case [closed]
1 A box contains 50 dices, half of them are fair, the others are not. Among the latter ones the face 1
appears with probability 1/2, while the other faces appear with probability 1/10. We randomly ...
-1
votes
1answer
24 views
Continuous random variables conditioned on discrete random variables
Suppose I have two continuous random variables $X_1 \sim N(0,\sigma_1^2)$ and $X_2 \sim N(0,\sigma_2^2)$
Then I have a discrete random variable $Z$ and $\theta$ such that
$$
\theta = \begin{cases}
...
0
votes
1answer
28 views
Let $Y$~$Gamma(a,b)$ with PDF $f(y)$= $y^{a-1}e^{-y/b}/\gamma(a)b^a$; $y>0$ For $a>1$, show that the mode of $Y$ is $(a-1)b$.
Suppose that $Y$ follows a gamma distribution with parameters $a$ and $b$. That
is,
$f(y)$= $y^{a-1}e^{-y/b}/\gamma(a)b^a$; $y>0$
For $a>1$, show that the mode of $Y$ is $(a-1)b$.
My working:
I ...
0
votes
1answer
43 views
Why should the support of a random variable be closed?
I am looking for an intuition regarding why the support of a random variable is closed.
I read for example that the support of an n-dimensional random variable as the "smallest closed subset of $...
2
votes
3answers
32 views
Let $X$~$U(0,5)$ & $Y$ be exponential random variable with with mean $2x$. Find the mean and variance of Y.
Let $X$~$U(0,5)$ & $Y$ be exponential random variable with with mean $2x$.
Find the mean and variance of Y.
My Working:
I know that the pdf of random variable $X$ is given by: $f_X(x)=1/5$
Since $...
-1
votes
0answers
72 views
Distribution of square root of positive normal random variables [closed]
Could you teach me what is pdf of square root of positive normal random variables if the distribution of normal variables follows
f(x)=1/(σ√2π) exp(-(x-μ)^2/(2σ^2 )), (x - μ)> 0
where μ and σ are ...
0
votes
0answers
24 views
How to calculate the following probability with two Markov chains?
I have two independent Markov chains at discrete time $X_{n}$ y $Y_{n}$, and I know the transition probabilities of both. I would like to know an easy way to calculate probabilities of the type: $$P(...
0
votes
3answers
109 views
$\mathbb{P(|\mathbf{X}|\leq 2) = 1}$ if $\mathbf{X}$ has bounded moments?
Suppose that a random variable $\mathbf{X}$ has bounded moments: $\mathbb{E}(\mathbf{X}^k)\le k^{2}2^k$. I would like to show that $\mathbb{P(|\mathbf{X}|\leq 2) = 1}$. I am considering using Markov's ...
3
votes
0answers
38 views
Expectation and correlation of eigenvector entries for random matrix
I'm looking for possible resources with results related to the theory of random matrices (this field is pretty far from my comfort zone, so I am not even sure where to start looking).
In a problem I ...
0
votes
0answers
17 views
Super-expectation of a random variable minus its quantile
Let $X$ be a continuous random variable, and denote $(X-c)^+ := \max(X-c, 0)$.
Let the superexpectation of X be given as
\begin{align}
E_{X}(c) := E[(X-c)^+],
\end{align}
as defined in
Rockafellar, ...
2
votes
1answer
45 views
$30\%$ students fail in Mathematics every year. If this year $500$ students take final exam, $P(145 \leq X \leq 400) = ?$.
It is given that $30\%$ students fail in Mathematics every year. If this year $500$ students take final exam then find $P(145 \leq X \leq 400)$. Here $X$ denotes the number of students fail in Maths.
...
0
votes
2answers
45 views
Distribution of Uniform Random Variable When Bounds Are Uniform
Let $X|_{Y=y} \sim U(-y,y)$ and let $Y\sim U(a, b),\space\space a,b\in\mathbb{R_{\ge0}},b>a$. What is the cumulative density function of $X$ when $Y$ is not known?
I know:
$$P(X|_{Y=y}\le x)=\frac{...
-1
votes
0answers
27 views
Random variable estimation stochastic [closed]
I am totally lost with this proof. Can someone please help me?
Show that if a random variable $X\geq 0$ satisfies, for some constants $c,C>0$ and $r\in\mathbb{R}$,
$P(|X-r|\geq t)\leq C\exp(-ct^2)$
...
0
votes
0answers
18 views
Expectation of random variable conditional on skewness [closed]
I'm interested in knowing the conditional expectation of a random variable X given it's skew.
Assume, I know X follows a log normal distribution with mean nu and variance sigma.
More formally in ...
2
votes
1answer
40 views
random variable, random sum
I have problem to show the following. Maybe someone has an idea :/
Let $X_1,...,X_n$ be i.i.d. copies of a random variable X with $|X|<1$. Let $S_n=X_1+..+X_n$. Then for any $A>0$:
$\mathbb{P}(|...
1
vote
0answers
17 views
rate of convergence of the a.s. zero series
Let us consider the following sequence $n^{s}X_{n}$, where $X_{n}$ are random variables defined on the same probability space, and $s>0$. Assume that we know that
$$
P\left(\liminf_{n\to\infty}A_n\...
