Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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1answer
15 views

Find and sample minimum of two exponential distribtions

I have two (or more) independent exponential variables $ X_1 \sim \exp(\lambda_1) $ and $ X_2 \sim \exp(\lambda_2) $. I want to get both the value of $ \min(X_1, X_2) $ and $ \arg\min(X_1, X_2) $. Can ...
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4answers
39 views

Difference between $E[X^2]$ and $E[X^3]$

Hope to ask a dumb question. $Y = aX$,with $a \in N_+$. Here, we know the correlation coefficient is 1. Now, suppose $X \sim N(0,1)$. Here, we know $X, Y$ are not independent. Cov($X,Y$) = ...
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0answers
20 views

Let $X_1,X_2\sim N(0,1)$. How to find joint pdf of $\,Y_1=X_1^2+X_2^2\,$ and$\,\,Y_2=\frac{\displaystyle X_1}{\displaystyle \sqrt{X_1^2+X_2^2}}$?

Let $X_1,X_2\sim N(0,1)$. How to find joint pdf of $\,Y_1=X_1^2+X_2^2\,$ and$\,\,Y_2=\frac{\displaystyle X_1}{\displaystyle \sqrt{X_1^2+X_2^2}}$? $$$$ I have tried to use Jacobian matrix to do ...
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0answers
14 views

How to compute the covariance matrix of a random variable uniformly distributed in an ellipsoid

Suppose that x is a random variable uniformly distributed in an ellipsoid \begin{equation} x^{T}Mx\leq\delta, \end{equation} where $x\in \mathbb{R}^{n}$. Clearly, the mean of $x$ is zero. The ...
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0answers
5 views

Just like Box Muller algorithm for random numbers in Gaussian distribution, are there any such algorithms for other distributions?

I want to create random numbers in various distributions like Poisson, Binomial, Gamma, etc. I cam across Box-Muller algorithm for random number generation in Gaussian distribution. Are there similar ...
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1answer
47 views

Integral $\int_0^\infty e^{-x/2}x\log(1+kx^2)\,dx$

How to evaluate: $$\int_0^\infty e^{-x/2}x\log(1+kx^2)\,dx$$ Basically am evaluating value of $\log(1+c\chi^2)$ where $\chi^2$ is $\chi$-squared distributed
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0answers
8 views

On Conditional distribution of the multivariate normal.

Following the answer to this question. Where we are talking about a multivariate normal than has mean and covariance matrix that can be decomposed as: $\boldsymbol\mu = \begin{bmatrix} ...
2
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1answer
44 views

Do not exist IID random variables $X, Y$ such that $X-Y \sim U[-1,1]$

This is an exercise from Williams, Proability with martingales. Prove that if $Z$ has the $U[-1,1]$ distribution, then $$\phi_Z(t) = \frac{\sin t}{t}$$ Then prove that do not exist IID random ...
2
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1answer
275 views

What is the continuous distribution version of multinomial distribution?

I am trying to model a distribution, on the number of occurrences of an event in a 24 hour time span. Right now, I discretize the 24 hour time span into hourly intervals, and each hour is taken as a ...
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0answers
14 views

A simple question about Delta Method's demonstration.

Suppose that $\sqrt{n}(X_n-\mu)\stackrel{D}{\longrightarrow}X$ and consider $g:\mathbb{R}\rightarrow\mathbb{R}$ a function such that first derivative $g'$ is continuous in a neighbourhood of $\mu$, ...
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1answer
269 views

How do you transform Gamma to Chi-squared distribution

Here is the question not sure how to turn a Gamma into a Chi-Squared: Suppose $X_1....X_n$ is a sample from the distribution Gamma($\alpha=3,\ \lambda=\theta$) with unknown $\theta > 0$. We wish ...
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1answer
106 views

Concentration of measure bounds for multivariate Gaussian distributions (fixed)

Let $\gamma_n$ denote the standard Gaussian measure on $\mathbb{R}^n$. It is known (see for example Cor 2.3 here: http://www.math.lsa.umich.edu/~barvinok/total710.pdf) that ...
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1answer
14 views

explanation of probability density function

How can we explain that if a random variable $X$ has pdf $f(x)$ then the function $Y=g(X)$ will have different pdf than $f(x)$ ?? And how to find the pdf of $Y=g(X)$ ??
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0answers
10 views

Calculation of probabilities in Z table

I would like to calculate at least one probability from z table. I know that pdf for N(0,1) is 1/(2*pi)*exp^(-(x^2/2)). Also the cdf is However, I do not know how to calculate this integral. ...
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0answers
10 views

sum of two dependent random variables

Let $X$ be a cotinuous random variable uniformly distributed over $[-10,10]$. Let $Y$ be a random variable with pdf $f_Y(y) = \frac{1}{40}\ln \frac{20}{|y|}, -20 \leq y \leq 20$. $X$ and $Y$ ARE NOT ...
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1answer
295 views

Mixture Gaussian distribution quantiles

Let $f_1(x), \dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, \dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = \sum_i w_i f_i(x)$ is also a ...
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2answers
15 views

Value of lambda in poisson distribution

I am currently studying statistical estimators and I came across a question that asks to give an estimate of the parameter λ of a Poisson distribution (using the method of moments), given that the ...
1
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0answers
8 views

transformation and functions of random variables

Let $X,Y$ be independent random variables. I already have the distribution of $XY$ over a certain subinterval of $\mathbb{R}$, by convolution. My question is, is it possible to get the distribution of ...
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0answers
24 views

Mean Preserving PDF Spreading

I have a histogram representing the PDF of an unknown discrete RV. The histogram is asymmetrical. To be clear, all I have is the histogram. Is there a known way to increase/decrease the variance of ...
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1answer
27 views

Summation of binomial number of poisson random variables

Z is summation of K random variables that each has Poisson distribution with different means. But, K is a Binomial random with parameters of n and p. I was wondering what is the distribution of Z?
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1answer
25 views

probability of X+Y which are two independent random variable & uniform distribution[0,1] [duplicate]

Two random variables X, Y are independent and both uniform-distributed in[0, 1]. How to calculate the probability density function Z=X+Y ? I tried below, $$f_X(x) = \begin{cases} \frac1{1-0} \\ ...
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0answers
27 views

Probability distributions with closed-form cumulative distribution functions (CDFs)

I am interested in finding multivariate probability distributions for which the cumulative distribution functions (CDFs) are given in close form. For instance, the multivariate Gaussian distribution ...
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0answers
139 views
+50

The Expectation of a function of independent random variables

Assume we have for some index $i>n$ ($n \in \mathbb{N} $) the following ${\it Independent \ Random \ Variables}$ $$h_i \sim \text {i.i.d }\ \ \mathcal{CN}(0,1) \ \ \text{ Complex Gaussian}$$ ...
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2answers
34 views

If x has a distribution function $F_x(x)$, what is the distribution function of $y = \exp(x)$?

I'm really struggling to figure out this problem from one of my practice exercises for a probability course. I know that the probability distribution function $f_x(x)$ is related to the cumulative ...
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1answer
23 views

Distribution of random variables when combined

I need help with this problem: If $X$ and $Y$ are two independent random variables and are both standard normal, what is the distribution of $\frac{1}{2}(X^2+Y^2)$? I think I start with ...
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0answers
16 views

Probability Formula for Posterior With 3 Variables

First post on math.stackexchange; pardon me if this is naive/a repeat. I'm following this document here by Prof. David M. Blei: ...
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2answers
33 views

Does strictly positive density function on the real line with infinite expected value exist?

The problem is as stated in the title. I am looking for an example or a disproof, whether there exists a continuous density function on the whole real line with infinite expected value. Once again: ...
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0answers
15 views

How do I prove that a given probability distribution is Gaussian

I am trying to plot the distribution of a random variable $x$. I got this distribution by marginalising a wishart distribution. When I plot the distribution curve of $x$, it looks like bell shaped ...
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1answer
25 views

Find conditional probability of random variables

I need to find conditional probability to count mutual information. Random variable X has uniform distribution on set ...
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0answers
18 views

Convergence of Uniformly Distributed Random Variables (n-dimensional)

Suppose that ${U_n} = ({U_{n1}},{U_{n2}},...,{U_{nn}})$ is uniformly distributed over the n-dimensional cube ${C_n}={[0,2]^n}$ for each $n=1,2,...$ That is, that the distribution of ${U_n}$ is ...
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1answer
1k views

Singular jacobian matrix?

I have a series of questions, in various degrees of befuddled muddledness (and they are related to my previous questions: this and this) First question: how do I do a change of variable if the ...
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0answers
8 views

Unknown bounded continuous distribution

Has the continuous distribution with the following probability density function in $(0,1)$ a name? $f(x;\alpha,\beta)=\frac{1}{\alpha^\beta\Gamma(\beta)}(-\log ...
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0answers
6 views

Conditional Probability in Multivariate Normal

Given a tri-variate Normal, the conditional probability of an element given others truncated information is Now if I know that the mean vector u is (-0.91,-1.31,-1.39) and R is ...
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0answers
20 views

sum/product combination of random variables

Let $X$ and $Y$ be independent random variables. If I am asked about the distribution of random variable $XY+Y$, is it ok if I compute $XY$ first and then add the result to $Y$ (via convolution, or ...
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0answers
11 views

Approximate a function with a gaussian distribution.

I have a function which has a bell-type graph and i need to find a Gaussian(Normal) with the appropriate mean, variance and constant factor which is close to the original function.The function in ...
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0answers
24 views

Probability density function definition

The definition above is given in my lecture notes. However there is no further reference/explanation given for what $o(h)$ represents. Can anyone explain this in this case?
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0answers
44 views

Expected value and Variance of a stochastic time integral of a deterministic variable (Standard Brownian motion)

Given a Standard Brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$, define: $$E(e^{\int_0^tudB_u})=?$$ $$ Var(e^{\int_0^tudB_u})=?$$ Sidenote to be edited later: Here is my try, I'm not capable to ...
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1answer
34 views

CDF of minimum of correlated and iid random variables

Consider two random variables $X_1=\min (W_1, W_2)$ and $ X_2=\min (W_3, W_4),$ where $W_1$, $W_2$,$W_3$ and $W_4$ are positive, identically distributed random variables. While $W_1$, $W_2$ are ...
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2answers
278 views

What is the probability that a student knows the answer given that he has answered it correctly,…?

A large class in stochastic processes at at a school is taking a multiple choice test. For one particular question with m proposed multiple choice answers, the fraction of students who know the answer ...
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0answers
19 views

Numerically stable way to compute the conditional covariance matrix

The Wikipedia article on multivariate normal distribution contains the well-known fact about the conditional "sub-distribution": If $μ$ and $Σ$ are partitioned as follows: $$ \boldsymbol\mu = ...
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0answers
16 views

Does martingale model work for betting football matches?

Imagine I have 1 million USD and will be betting 1.000 USD on the win of FC Barcelona each time they play a match in La Liga (Spanish Tier 1 football league). If FC Barcelona loses or ties their last ...
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1answer
624 views

Find unknown value in probability density function

"Suppose that a random variable $Y$ has a p.d.f. given by $f (y) = ky^3*e^{-y/2}$ when $y > 0$, and otherwise 0. Find the value of $k$ that makes $f (y)$ a density function." I found that $k=1.$ ...
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1answer
23 views

Moment generating function gives an undefined first moment, but first moment still exists?

Let's say we have a probability density function given by $f_X(x) = 2x$ for $0 \leq x \leq 1$. (Note $\int_0^1 f_X(x) = 1$.) The moment generating function is $$\int_0^1 e^{tx}\cdot2x \,dx$$ ...
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0answers
22 views

Determine the distribution of the random variable [on hold]

The number of chimney fires in a large city over a week with an average of about 520 fires annually being blames on fireplaces, chimneys or chimney connectors.
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0answers
12 views

log-concavity with PDF and CDF

Assume the following: pdf: $f_X(x)$ cdf: $F_X(x)=P(X \leq x)$ $X$ is a random variable with log-concave pdf $f_X(x)$. $Y = h(X)$ $X \in R^n$ $h: R^n \rightarrow R$ Through the ...
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0answers
9 views

sum of two Gaussian random variables conditioned on their sum

I have two independent standard normal R.V.s X and Y, and their sum is Z = X + Y. I am trying to calculate the PDF of X conditioned on Z taking the value z. I know that this is the joint PDF of X and ...
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0answers
23 views

Probability (Please see picture below) [closed]

On a scratch card you win if you find a sun in the first square you scratch off. Here are the scratch cards before the suns are covered. $$\mathbf{A}\ \ \ \ \ \ \ ...
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1answer
19 views

Computation of two-sided probability density functions from their cumulants using Laplace transform

The computation of one-sided probability density functions (PDFs) from their cumulants using Laplace transform is proposed by following paper: M.N. Berberan-Santos, Journal of Mathematical Chemistry, ...
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1answer
8 views

Expectation and variance of X − Y

Let's say I have $X=\min\{X_1,...,X_{10}\}$ with the $X_i\sim Exp(\lambda_i)$ independent. And let $Y=\min\{X_{11},...,X_{20}\}$ What is the expectation and variance of $X-Y$? I really don't know ...
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0answers
10 views

Given the sum of four Exp(1) distributed random variables, what is the conditional density of sum two of them?

Let T := X+Y+Z+K be indepedent and Exp($1$)- distributed random variables. What is the density of (X+Y) given {T = $1$} ? For M:= X+Y and N := Z+K given {M + N = $1$} The joint density is $ ...