Tagged Questions

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

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1
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1answer
23 views

problem on random variable in probability

A game consists of first rolling an ordinary 6-sided die once and then tossing a fair coin once. The score, which consist of adding the number of spots showing on the die to the number of heads ...
-5
votes
0answers
24 views

To Find Covariance Matrix From a Random Variable Matrix [on hold]

$$X=\begin{pmatrix} 1& 2& 3\\ 2& 4& 1\\ 3& 1& 1\\ 4& 1& 2 \end{pmatrix}$$ What is the covariance matrix? Please Help.
0
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0answers
17 views

Determining if matrix could be a valid covariance and cross-covariance matrix

Given a matrix, $M$, how can you determine the following? Could it be a valid covariance matrix of some random vector? Could it be a valid cross-covariance matrix of two random vectors? Could it be ...
0
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0answers
50 views

conditional random variable?

Can we really talk about conditional random variable (X$\mid$B); a random variable defined by the conditional distribution of the random variable X given the event B, or this is completely wrong? ...
0
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1answer
28 views

Given density $f_U(u)$ and $U=X_1/X_2$, can we find $f_{X_1}(x_1)$ and $f_{X_2}(x_2)$?

Problem Statement: Let $X_1$ and $X_2$ be independent random variables. Given the density function, $ f_U(u)$, and the relationship: $$U=X_1/X_2$$ Can we find the density functions for $X_1$ and ...
0
votes
1answer
18 views

Entropy of geometric random variable?

I am wondering how to derive the entropy of a geometric random variable? Or where I can find some proof/derivation? I tried to search online, but seems not much resources is available. Here is the ...
0
votes
1answer
33 views

Given a random value and its expected value, how can you determine if another random variable exists?

Suppose there exists a random variable $X$ with $E[X] = 1$, can $E[2^{-X}] = \frac{1}{4}$ exist? Can $E[2^X] = 8$ exist? Is there a general method to solve this type of problem?
1
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0answers
20 views

Division of Dependent Random Variables

Let $X_1$ and $X_2$ be dependent random variables. Find the density function for: $$U=X_1/X_2$$ Can I use the transformation method to find the conditional density and then use method of ...
1
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1answer
38 views

Proof of $E(X)=a$ when $a$ is a point of symmetry

I am trying to develop a proof of the following: Given a random variable $X$ with symmetric probability density function $f(x)$, prove that $E(X)=a$ where $a$ is the point of symmetry. A couple of ...
2
votes
1answer
59 views

Weak form of Berry-Esséen theorem

Let $X$ (a real random variable) have mean zero, unit variance and finite third moment. Let $Z_{n}:=(X_{1}+...+X_{n})/\sqrt{n}$, where $X_{1}, ... X_{n}$ are iid copies of $X$. According to the ...
0
votes
1answer
21 views

Showing $\int_0^{\infty}(1-F_X(x))dx=E(X)$ in both discrete and continuous cases

Ok, according to some notes I have, the following is true for a random variable $X$ that can only take on positive values, i.e $P(X<0=0)$ $\int_0^{\infty}(1-F_X(x))dx=\int_0^{\infty}P(X>x)dx$ ...
2
votes
1answer
20 views

Question regard the notion of almost sure convergence

Consider an $n\times m$ matrix with i.i.d. entries each having zero mean and variance $1/n$. Let $Y = X^TX$. By the strong law of large numbers, we know that the $(i,j)$ entry of $Y$ goes almost ...
0
votes
1answer
44 views

Density of the sum of two random variables with parameters.

I've got a problem from probability theory, which is supposed to have simple theoretical solution. The statement is: "Suggest a probability distribution $(ζ,η)\inℝ^2$, such that random variable $γ = ...
0
votes
1answer
51 views

Given a function and how do you determine the pdf of the left side given the pdf of the right side variables?

Given a function and how do you determine the pdf of the left side given the pdf of the right side variables? Specifically what is the pdf of W, given the equation $$ W = I^2 R$$ with $I$ and $R$ are ...
1
vote
0answers
42 views

Calculate the variance from a function of normal random variable

I am new to the topic that I found difficulty for the question: Given the function $g(x) = e^{-X}$, $X \sim N(0,1)$, calculate the variance of $g(x)$. I know the answer is $e(e-1)$. But I don't ...
0
votes
1answer
38 views

Joint pdf of independent randomly uniform variables

Given two random uniform variables, $U$ and $V$, that are uniformly distributed over [0,1], how do you calculate the joint pdf of $X$, $Y$ where $X = F(U,V)$ and $Y = G(U,V)$ and where is the joint ...
3
votes
2answers
135 views

Sums of Products of Two Normal Variables

Suppose that $X_1 ,\ldots,X_n,Y_1,\ldots,Y_n$ are all independent normal random variables with different means and variances. What is the PDF of the following random variable? ...
1
vote
1answer
29 views

Can the sum of two pregaussian random variables be zero?

Do there exist any two Sub-Gaussian random variables with variance 1 such that the sum of them is the $0$ random variable, whose value is $0$ with probability $1$?
0
votes
1answer
52 views

Probability of a random variable dependent on a parameter.

Let $X_L$ be a random variable dependent on a parameter $L$, taking only discrete values between $0$ and $+\infty$. Let $\mu L$ be its expectation, where $\mu$ is a costant. Which conditions should I ...
0
votes
2answers
68 views

What is the distribution of $c^x$? ($c$ is a constant, $x$ is a random variable)

What is the distribution of $c^x$, where $c$ is a constant and $x$ is a random variable? For example, $x$ follows a Poisson distribution, what is the distribution of $2^x$?
1
vote
0answers
25 views

Laplace transform of a sum of stochastic variables

I have a problem with interpretation of one transformation performed on equation consisting of continuous random variables. Here is the source equation describing recurent relationship between the ...
1
vote
1answer
9 views

nonlinear transform of Gaussian random variable that preserves Gaussianity

I recently know that following results. suppose that $x_1, x_2, x_3$ are independent real Gaussian random variables with $\mathcal{N}(0, 1)$. Then $$ \frac{x_1 + x_2 x_3}{\sqrt{1+x_3^2}} \sim ...
0
votes
2answers
56 views

Fisher's information for two independent random variables

If $X$ and $Y$ are two independent random variables, with regular distributions, how can I prove $I_{x,y}(\theta) = I_x(\theta) + I_y(\theta)$ ? Thanks! I tried: $$ {\rm E}_\theta ...
2
votes
2answers
68 views

Ergodicity of a sequence of independent blocks

I am stuck with the problem given below, more precisely, with the part regarding ergodicity. I have a proof, also given in what follow, but it does not seem to be correct; well, at least, it does not ...
1
vote
1answer
24 views

Correlation of sums of correlated variables

I'm trying to work out an expression for a correlation of the weighted sums of two r.v.'s with a third r.v. To be precise, I have a trivariate normal distribution: $$\{X,Y,Z\}\approx ...
0
votes
0answers
45 views

If $X$ is a random variable, is $p( X )$ also a random variable?

If $X$ is a random variable with density/mass function $p$, is $p(X)$ also a random variable?
3
votes
2answers
56 views

PDF of summation of independent random variables with different mean and variances

What can we say about the probability distribution function of $n$ independent random variables with different means and variances but the same PDF? For example, lets say $X_1,X_2,\dots,X_n$ are ...
0
votes
2answers
41 views

Prove that the CDF of a random variable is always right-continuous

Let $X$ be a random variable with cumulative distribution function $F_X$. It is a known fact that this function $F_X$ is right-continuous. But I'm having some trouble to prove this result. Below I'm ...
0
votes
0answers
39 views

Generalized Rayleigh distribution?

A variable $z=\sqrt{x^2+y^2}$ is Rayleigh distributed with parameter $\sigma$ if $x\sim\mathcal{N}(0,\sigma^2)$ and $y\sim\mathcal{N}(0,\sigma^2)$. Is there a known/named distribution for the case ...
2
votes
1answer
51 views

Basu's Theorem's application

I can't solve a problem, for which I am suppose to use Basu's theorem. Suppose that $X$ and $Y$ are independent Exponential random variables with common parameter $\lambda$. I have to show that $X + ...
0
votes
1answer
66 views

probability density and distribution

Let $X$ and $Y$ be uniform random variables where $0\leqslant x,y\leqslant 1$. Let $\operatorname{sgn}(x)$ be $1$ when $x>0$, $−1$ when $x<0$ and $0$ when $x=0$. Find the distribution and ...
2
votes
1answer
20 views

Discovering the joint distribution for two dependant RVs?

Suppose I have three continuous random variables. X1, X2, and Y, where Y = X1+X2, and X1 and X2 are dependent. If I know the probability distributions separately for X1, X2, and Y, is there a ...
2
votes
2answers
82 views

Almost sure convergence of a sum of random variables

Suppose $(X_i)_{i=1}^{\infty}$ is an i.i.d. sequence of rv's, where $X_i$ can take countably many values $\{x_1,x_2,\dots\}$ with probabilities $\{p_1,p_2\dots\}$, respectively. Let $p_{n,k}:= ...
2
votes
1answer
39 views

Prove expectation inequality

Any ideas on how I could prove the veracity or falseness of the following inequality? Let $X:\Omega \to \mathbb{R}$ a random variable such that the expressions under are well-defined. Then $$E[e^X] ...
0
votes
0answers
64 views

Measurable random variable with respect to a finite partition

Let $(\Omega,\mathcal F,P)$ be a probability space, and $X$ is a random variable on $\Omega$. If $\mathcal G$ is a sub sigma algebra of $\mathcal F$, generated by a partition $\{ A(i),\; ...
1
vote
1answer
36 views

Issue with a Poisson process and its jump times

Let $(N_t)_{t\geq 0}$ be a Poisson process and $$T_n = \inf\{t\geq 0, \ N_t \geq n\}$$ Now given $t \ge 0$ how to compute $$ \mathbb{E} \left[ \sum_{n=1}^{N_t} X_{T_n}\right] $$ ? where $(X_t)_{t\ge ...
-1
votes
1answer
24 views

Expectation of an Inverse of a Random Variable

$X$ and $Y$ are two discrete random variables taking values greater than 1. Given, $\mathbf{E}[X] \geq \mathbf{E}[Y]$, either prove that $\mathbf{E}[\frac{1}{X}] \leq \mathbf{E}[\frac{1}{Y}]$ or ...
1
vote
1answer
34 views

What does orthogonal random variables mean?

As far as I know orthogonality is a linear algebraic concept, where for for a 2D or 3D case if the vectors are perpendicular we say they are orthogonal. Even it is OK for higher dimensions. But when ...
0
votes
1answer
23 views

Law of Gaussian Vector and Matrix

I have a probability problem that I’m struggling with. Any help would be appreciated. I have no clue about how to solve this. Here is the problem: Let A be: \begin{pmatrix} 4 & 7 \\ 1 & -2 ...
3
votes
1answer
44 views

Criterion for independency of random variables

I saw in some notes the following "criterion" for independency of two random variables. Let $X$ and $Y$ be real-valued random variables defined on the same space. $X$ and $Y$ are statistically ...
1
vote
0answers
21 views

Derive Student T distribution using transformation theorem

I am trying working on an exercise that asks me to show that If $ X_1 \in N(0,1) $ and $ X_2 \in \chi^2(n) $ are independent random variables, then $ X_1 / \sqrt{X_2/n} \in t(n) \, $ where $ ...
3
votes
1answer
37 views

Distribution of the minimum of two random variables

Here is my problem. Let us consider X, Y and Z to be random variables following the exponential distribution (same mean or not does not matter). I am trying to find the distribution of the ...
0
votes
0answers
37 views

Transforming 2 Random Variables from Joint Uniform Distribution using Jacobian

I am trying to transform 2 random variables A and B to 2 other variables X and Y and get the joint probability density function of X and Y. I am given the equations: A = ln[Y / (1-Y)] and B = ...
2
votes
1answer
72 views

Random processes: Repair time

I have a question that is to do with qeueing theory and repair times: Assume that a small office has 4 printers. Each printer breaks down independently of the other printers and independently of the ...
3
votes
1answer
93 views

Weak convergence in the Skorohod space $D([0, T])$ $\forall T$ implies Weak convergence in $D([0, \infty))$?

Assume that $Z_n$ are random variables taking value in the Skorohod space $D([0, \infty),Y)$ (endowed with its usual Skorohod topology) of right-continuous functions $[0, \infty) \to Y$, where $Y$ is ...
0
votes
1answer
34 views

Conditional probability over a function

I have a question if the following relations on conditional probabilities hold for independent random variables? $$P_{X \mid Y, G(Y)}(x_1)=P_{X \mid \{Y\}}(x_2)$$ where $G$ is not necessarily ...
1
vote
1answer
56 views

distribution of maximum of $n$ Pearson correlations

$\mathbf{x}=[x_1,x_2,...,x_m]^{\top}$ is a vector of length $m$ and $\mathbf{y_1}, \mathbf{y_2}, ..., \mathbf{y_n}$ are similarly $n$ vectors of length $m$. If the elements of $\mathbf{x}$ and ...
0
votes
1answer
48 views

Conditional distribution of a function of random variables

I have a question about conditional distribution. Suppose we have three independent random variables $X_1$, $X_2$, $X_3$. Then we have mapping $Y_1=g(X_1, X_2)$. The mapping is not necessarily an ...
1
vote
2answers
22 views

Results about conditional expectations

$\theta$, $\phi$ are integrable random variables on a probability space $(\Omega,\mathcal{F},P)$ and $\mathcal{G}$ is $\sigma$-field on $\Omega$ contained in $\mathcal{F}$. Now we want to prove ...
0
votes
1answer
27 views

How do you calculate the expected value, the variance and the standard deviation of a sum of random variables, regardless of their distribution

I was looking for a formula or something, but I can't find anything anywhere. So, can someone tell me the steps I need to follow in order to do that? I thought that the expected value of the sum was ...

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