Questions tagged [random-variables]

Questions about maps from a probability space to a measure space which are measurable.

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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 $a + bX + c X^2$ ...
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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 $(\Omega, \mathcal{F},P)$ ...
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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
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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 ...
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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 $500$ ...
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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}...
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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 ...
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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 ...
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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 $P$ ...
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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 ...
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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 ...
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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 ...
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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} \...
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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 ...
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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}{\...
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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 ...
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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 ...
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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 ...
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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 $...
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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 ...
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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(...
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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 ...
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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^...
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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 ...
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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 : \{ ...
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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 ...
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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 $...
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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 ...
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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 ...
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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'...
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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 $...
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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 ...
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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 $h(x,y)=2x+5y$...
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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 ...
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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 ...
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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} ...
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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 ...
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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 $...
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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 $...
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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 ...
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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(...
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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 ...
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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 ...
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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, ...
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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. ...
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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{...
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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 $r\in\mathbb{R}$, $P(|X-r|\geq t)\leq C\exp(-ct^2)$ ...
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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 ...
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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}(|...
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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\...

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