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Questions tagged [random-variables]

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

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2answers
16 views

About Conditional Variance $X$ Has distribution $ U(0,1)$ and $Y$ has distribution $ U(0,X)$

I have a question about the problem mentioned above, the main says $X$ Has distribution $ U(0,1)$ and $Y$ has distribution $ U(0,X)$ Find $E(Y)$ and $Var(Y)$ I try to take it for $E[Y|X]=X$ and $...
2
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0answers
18 views

Distribution of sum of random phases

I want to find the distribution of the following random variable: $v = \| a_1b_1e^{j\theta_1} + a_2b_2e^{j\theta_2} + \dots + a_Mb_Me^{j\theta_M}\|$ where $\theta_i$ is a random variable with known ...
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1answer
26 views

PDF transformation for many to one function

I would like to find the PDF of the random variable $Y$ given the PDF of $x$. $$Y=sin(x)$$ $$f(x) = 2x/(pi^2) for 0<x<pi$$ and 0 otherwise. Following the tips in the question here: ...
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1answer
15 views

Expected value and modulo

Let $X \in \mathbb{N}$ be some discrete RV and define $Y = X \mod k$. The value of $Y$ is the representative in the coset of $X \mod k$ in $[0, k-1]$. For example if $X = 9$ and $k = 4$ then $Y = 1$...
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0answers
13 views

Is an exponential function of a stochastic process a smooth function?

Let's say I have an Ito process $X_t$, and another process $Y_t=e^{\int_t^{t+\delta} X_s ds}$. I want to know that quadratic variation of $Y_t$ and another process. I know that the quadratic ...
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0answers
10 views

Gauss-Markov Mobility Model query

I have a query in understanding a little point, please if you could help me understand that. Thanks In the Gauss-Markov Mobility Model, the current speed and position are calculated based on speed ...
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0answers
20 views

Distribution of exp with Rayleigh arg

I am wondering what the distribution of $$ y = exp(i\phi)$$ where $\phi$ is Rayleigh distributed and $i=\sqrt{-1}$. My thought process to solve was to use LOTUS but I am unsure how to handle the $...
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2answers
29 views

How can I interpret the variance of a random variable?

Let $X$ a random variable. I know how to calculate it's variance but I don't really understand how to interpret it. For example, if $(B_t)$ is a Brownian motion $B_t\in \mathcal N(0,\sigma ^2t)$. How ...
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0answers
12 views

Find the probability of error given two event condition to random variable?

Let $X_1,X_2$ be random variables with PDF $f_{X_1}(x)$ and $f_{X_2}(x)$ receptively . The probability of error $P(e)$ condition to $X_i$ is given by $$ P(e|X_i)=F(X_i) $$ where $F(x)$ is a function....
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1answer
37 views

With regard to random variables, does $(X/Y)$ independent of $(Y)$ imply that $(X)$ is independent of $(Y)$?

This makes logical sense to me, but I can't seem to prove this. Is this statement true? Note: X/Y is a ratio here, not conditioning.
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1answer
9 views

Tail weight of product distributions

Are there any general results relating the tail weight of two (or more) probability distributions to the tail weight of their product distribution (in particular, on the assumption that the ...
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0answers
28 views

How to understand such a random variable from the perspective of probability theory?

I just came across such a situation: Consider such a space that glues 3 $\mathbb{R}_{\geq 0}$ together at 0, we call it $S$. That means the image of the random variable from the sample space $\Omega$ ...
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0answers
15 views

Notation for expectation of a random variable/general Lebesgue integral

Suppose $(\Omega, \mathcal{F}, \mu)$ is a probability space and suppose I have a random variable $X: \Omega \rightarrow \mathbb{R}$. By definition, we have $$\mathbb{E}[X]=\int_{\Omega}Xd\mu=\int_{\...
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1answer
28 views

On the distribution of a normalized Gaussian vector

Let $x=(x_1,\ldots,x_n)\in\mathbb{R}^n$ be an $n$-dimensional random vector that follows the normal distribution with mean vector $\mu$ and covariance matrix $\Sigma=\operatorname{diag}\left(\sigma_1^...
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0answers
24 views

Show that T achieves the Cramer Rao lower bound

Problem Statement: Consider $T$ to be an estimator of $\theta$. Show that $T$ achieves the Cramer Rao lower bound if and only if $Z$ is a linear function of $T$ $Z=a(\theta)T+b(\theta)$ ...
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1answer
31 views

Minimum of two random variables with exponential distribution

Let $X,Y$ be two random variables with exponential distribution and their rates are $\gamma, \beta $. Let $Z$ be a random variable such that $Z = min\{X, Y\}$. How do I prove that the density ...
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0answers
12 views

Terminology for dependent random variables

Let $A(\omega)$ be a random variable with a uniform distribution over $[-1, 1]$. Let $B(x, \omega)$ be a random process that is a function of physical space $x$. It is defined in terms of $A$ such ...
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50 views
+50

Prove that $\frac{1}{n}D_{n}\to \frac{\pi}{4}$ a.s.

Let $X^{n}:=(X_{1}^{n},X_{2}^{n})$ and $(X^{n})_{n}$ be IID random variables where $X^{n}$~$\mathcal{U}(K)$ on a probability space $(\mathbb R^{2}, \mathcal{B}(\mathbb R^{2}), P)$ where $\forall A \in ...
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1answer
12 views

$Y = \frac { K A ^ { 3 } } { ( B + D ) ( C - D ) }$

K is a constant Find an expression to approximately determine the variance of Y, assuming $A , B , C ,$ and $D$ are probabilistically independent. isnt the expression they have already given me the ...
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2answers
26 views

When X,Y are independent random variables we can use convolution to find the density of $X+Y$, can we do that for $X-Y$? [on hold]

When X,Y are independent random variables we can use convolution to find the density of $X+Y$, can we do that for $X-Y$? Is there an analogical case?
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1answer
15 views

Proof of $E[\min{(X,y)}]=\int\limits_{x=0}^{y}{(1-F(x))}\text{d}x$

I'm trying to prove the following theorem. Theorem. If $X$ is a positive random variable with distribution function $F$ and density function $f$, then, for each $y \geq 0$, $$E[\min{(X,y)}]=\int_{x=...
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0answers
13 views

Compute E(p) where p is a mixed random variable

Problem Statement: Suppose that $p$ is a mixed random variable with the discrete part having pmf $ Pr(p) = \left\{ \begin{array}{lr} 0.1 , & p = 1/3\\ 0.2, & p=1/2\\ 0....
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0answers
20 views

It is not necessarily true that if $X,Y$ are continuous then $(X,Y)$, but if they are independent?

I have a general questions. It is easy to disprove that for any X,Y continuous random variables then $(X,Y)$ is continuous. But what about if they are independent? Then I can define a new ...
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0answers
23 views

Find PDF of $(X-Y)^2$- uniform distribution.

Let $X,Y \sim U[-1,1]$. $X$ and $Y$ are independent.How to find pdf of $Z=(X-Y)^2$? My idea: $P(Z \le t)=P((X-Y)^2 \le t)=P(-\sqrt{t}\le X-Y \le \sqrt{t})=P(X-Y \le \sqrt{t})-P(X-Y \le -\sqrt{t})=...
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1answer
13 views

Sum of Distribution Functions

Given $X$ follows distribution function $F$, $Y$ follows $G$ and $Z$ follows $H$ Express $Z=f(X,Y)$. Say $H=\lambda F+(1-\lambda)G\,\,\,\,(\lambda\in[0,1])$ Then find $f$ I was able to verify that $H$...
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1answer
25 views

Are the projections along orthogonal direction of multivariate normal distribution with diagonal covariance matrix independent?

I'm taking a probability class and my prof used the following theorem IIRC. Let $g\sim\mathcal{N}(\mu,\Sigma)$ where $\Sigma$ is diagonal( I don't know if this condition is necessary) and $\langle u,...
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0answers
26 views

Show that X (not conditioned on p) is a Bernoulli random variable [closed]

Problem Statement: Suppose that $p ∼ U[0, 1)$. Show that $X$ (not conditioned on the value of $p$), is a Bernoulli random variable with pmf $ P(x) = \left\{ \begin{array}{lr} E(p) , &...
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1answer
29 views

Property of median of probability distributiom

Suppose that a random variable $\mathbb{X}$ has density $f$ and a unique median $m$ . Suppose that $b$ is any real number. Show that $\mathbb{E(|X − b|) = E(|X − m|) + 2 \int_ b^ m (b − x)f(x)dx}$ , ...
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1answer
21 views

Trying to find the CDF of $X+Y$ when $X\sim exp(\alpha)$ and $Y \sim exp(\beta)$ (independent) without convolution, but it doesn't seem to work

The textbook I am using, using convolution in order to find the CDF of the $X+Y$ when $X\sim exp(\alpha)$ $Y\sim exp(\beta)$, and X and Y are are independent. However, I have no background with ...
1
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1answer
26 views

Convergence of the sum of iid scaled by $n^\alpha$

I am interested in the convergence of the sequence $\mathbb{P}(|X_1+...+X_n|/n^\alpha<z)$ where $z>0$, $\{X_n\}_n$ is an i.i.d. sequence with mean zero and finite variance. I can easily prove ...
0
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1answer
32 views

Is Supremum of sequence of bounded random variables a random variable?

I was thinking about the following situation : Suppose we have ${\{X_n\}}$ is a sequence of bounded random variables . Is it true that $\mathbb{lim \ sup} X_n$ is also a random variable ? ( I get a ...
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1answer
24 views

Is my computation correct? distribution of an inverse function

Let $ X \sim U[-1,2]$ and let Y: $$ Y=\begin{cases} \frac{1}{X^2} && X \ne 0 \\ 0 && X=0 \end{cases} $$ Find the distribution of Y. What I did was: $F_Y(t)=P(Y\leq t)=P(\...
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0answers
13 views

Transformation of an exponential RV [closed]

Suppose $f(y)=\theta exp(-\theta y)$; let $x=y^2$. Find $f(x)$. Thanks.
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0answers
11 views

Prove convergence of a sum of random variables

I am trying to grab on to some intuition about the area where random variables start looking a bit more like calculus. I've learned about random variables and the weak law of large numbers, but seem ...
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0answers
23 views

Setting up expression for distance of random point to origin

Problem Statement: Suppose that the coordinates of a point $(X, Y)$ are such that $X ∼ U[0, 1)$ and $Y ∼ U[0, 1)$ Compute the probability that the distance from this point to the origin is ...
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2answers
71 views

Show that $\mathbb E[X_{n}]\xrightarrow{n \to \infty} \infty$ while $X_{n} \xrightarrow{n \to \infty} 0$ a.s.

Say I have a biased coin that shows heads with probability $p \in ]1/3,1/2[$ and I initially have capital of $100 $EUR. Every time heads is shown, my capital is doubled, in the other case I pay half ...
3
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2answers
31 views

$X \sim \exp(1)$, $Y \sim \exp(1)$ Independent what is the CDF of $Z=X-Y$?

Let $X \sim \exp(1)$, $Y \sim \exp(1)$ $Z=X-Y$. $X,Y$ are independent. What is the distribution of Z? For $t\geq0$, I simply calculated that the old fashioned way. $F_Y(t)=P(Z\leq t)=P(X-Y\leq t)=\...
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1answer
36 views

A simple probability problem - not sure about my computation

Here's a pretty simple problem: A factory produces 3 types of lamps, A, B, C. 30% of the lamps are of type A, 30% B and the rest 40% C. The lifetime of each type is distributed exponentially, ...
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1answer
47 views

$2\int_{0}^{\sqrt{t}}\frac{1}{\sqrt{8\pi}}e^{-\frac{x^2}{8}}dx=\int_{0}^{t}\frac{1}{\sqrt{8\pi}}e^{-\frac{y}{8}}\sqrt{y}dy$?

I encountered the following "claim" in a probability exercise solution (find the distribution of $Y=X^2$ where $X\sim N(0,2)$ $$2\int_{0}^{\sqrt{t}}\frac{1}{\sqrt{8\pi}}e^{-\frac{x^2}{8}}dx=\int_{0}^{...
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0answers
30 views

Probability and random discrete variables: the PMF for the number of rolls it takes for a 6 sided die to repeat a number

We keep rolling the die as long as no value is repeated. When we see the first repeated value, that is the last roll. Let $X$ be the number of rolls it took. What is the PMF of X. Here's what I came ...
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0answers
33 views

Finding the CDF of a Sum of I.I.D Continuous Random Variables

Let X$_1$ and X$_2$ be identically independent distributions(i.i.d) random variables with $$\Bbb P(X_i \le x) = 1-x^{-1/2}, \quad x \ge 1 \ \text{and} \ i = 1,2 $$ Find $\Bbb P(X_1 + X_2 \le x)$. I ...
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1answer
30 views

Find $c \geq 0$ so that $c\hat{\vartheta}$ is unbiased

I have found the following statistical model. Say $\Omega:=[0,\infty)^{n}$ and $P_{m}$~$[\mathcal{U}(0,m)]^{\otimes n}$ where $m$ is the parameter $\in [0,\infty)$ and $n \in \mathbb N$ Define the ...
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2answers
23 views

A random variable $X$ is number of boys out of $n$ children. Calculate $\operatorname{Var}(2X-n)$

Let a random variable $X$ be the number of boys out of $n$ children. The probability to have a boy or a girl is $0.5$. Calculate $V(2X-n)$. I know that $Var(2X-n)=4V(X)$. $\mathbb{P}(X=k)={1\over 2^...
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0answers
42 views

Expected value of symmetric random variables

A random variable $X$ is symmetric if $X$ has the same probability distribution as $-X$. In the discrete case symmetry means that $P(X = k) = P(X = -k)$ for all possible values of $k$. In the ...
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2answers
37 views

probability question on casting a pair of dice

A pair of dice is cast until a seven appears twice or until each of a six and eight has appeared at least once. Show that the probability of the six and eight occurring before two sevens is 0.546. An ...
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0answers
17 views

Modelling Random Variables with Specified PDF and Correlation

I am trying to develop a radar simulation system that is able to generate random processes whose elements are taken from a specified probability density function and have also have a specified ...
2
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2answers
24 views

Equivalence of the sum of random variables and their expectation

Given that $X$ is a random variable I define $$ \psi = \sum_i^{n} X_i $$ so $\psi$ is the sum of $n$ variables with the same distribution. Given that $$ \bar{X} = \frac{\sum_{i}^{n} X_i}{n} $$ I ...
0
votes
1answer
29 views

Random Multinomial Variable

Nine persons go into a 3-carriage tram. Each person chooses the carriage at random. Which are the probabilities of the event: A : “There are 4 person in a carriage, 3 in another, and 2 in the other ...
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0answers
9 views

Ordinal scrambling quantification upon placing and retrieving labeled spheres to and from a cylindrical container.

Trying to derive a formula to quantify the degree to which objects in an original order are scrambled upon some amount of repeated random handling. Say there are $n$ spheres labeled with labels $1$ ...
0
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0answers
36 views

Question on $\liminf A_{n}$ and $\limsup A_{n}$

Say I have the following event $\{\limsup_{n \to \infty} |X_{n}|> \epsilon\}$, and that $P(\{\limsup_{n \to \infty} |X_{n}|> \epsilon\})$, I also realize that $\{\limsup_{n \to \infty} |X_{n}|&...