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

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2k views

Product and division of gamma distributions

SHORT VERSION: how would I go about solving the following inverse problem: $F(a,a_v,b,b_v,c,c_v)=\frac{A B}{C}$ where $A,B,C$ are gamma distributed according to A's mean=$a$, A's variance=$a_v$, etc. ...
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1answer
66 views

Is there a name for this family of probability distributions?

I am wondering whether a family of probability distributions with the following form of a density function has a name: $$f(x)=C*\operatorname{Exp}(-B|x|^A)$$ where $A$, $B$ and $C$ are positive ...
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2answers
230 views

How do I calculate the limit of this integral?

Using an appropriate probability distribution or otherwise show that $$\lim_{n\to\infty} \int_0^n e^{-x}{x^{n-1}\over(n-1)!}dx =0.5$$
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2answers
142 views

Are the square and the maximum of distribution functions a distribution function?

Let $F$ and $G$ be (one dimensional) distribution functions. Decide which of the following are distribution functions. (a) $F^2$, (b) $H$, where $H(t) = \max \{F(t),G(t)\}$. ...
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2answers
1k views

Singular Distribution

In reading section 2.2, page 14 of this book, I came across the term "singular distribution". Apparently, a multivariate Gaussian distribution is singular if and only if it's covariance matrix is ...
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2answers
44 views

How can I do a constructive proof of this:

Say Z is a non-negative R.V, and P(Z>0)>0. Then exists a a>0 and an b>0 such P(Z>a)>b. I am not sure how to start with the proof, I've been assigning numbers than can qualify for some CDFs but I don´t ...
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1answer
269 views

Why is this multivariate $3\sigma$ ellipse rotated?

While reading this answer, I clicked on the provided link to this Wikipedia page. The main article image shows the PDF of a 2D multivariate normally distributed system: In the image, the $3\sigma$ ...
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2answers
158 views

Conditional expectation of indicator function

Could someone confirm if the following is correct. If not why? \begin{equation} E[\mathbb{1_{X\leq x}}|Y]=P[X|Y]=\frac{P[X,Y]}{P[Y]} \end{equation} Thank you.
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67 views

Joint probability generating functions, help please!

With a sequence of $N$ independent Bernoulli trials performed, where $N \in \mathbb{Z}^+$ and the probability of success on any trial is $p$, and $S$ and $F$ being total number of success and fails ...
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1answer
141 views

Probability given X & Y are independent rand. variables and 2 p.d.f.s

first time poster here, maybe you guys can help me out. Given that $X,Y$ are independent random vars with these pdfs: $$f_X(x) = \begin{cases}1,& 0 < x < 1,\\0,& ...
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1answer
49 views

Tail of a sequence of RV

Consider the sequence of random variables $\{X_n\}_{n=1,2,\dotsc}$ where $X_n$ is gamma-distributed with shape $n$ and scale $1/n$ (or equivalently, $2nX_n$ is $\chi^2$-distributed with $2n$ degrees ...
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2answers
812 views

Convolution of continuous and discrete distributions

Assume we have two random variables $X$ and $Y$, such that $X \sim P(x)$ and $Y \sim G(y)$. We ask, what is the distribution of $Z = X+Y$. If both of the distributions of $X$ and $Y$ are discrete, ...
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1answer
137 views

Normal random Vector

Question: Prove that linear functions of the form $\bar{y}=\bar{b}+\mathrm{B}\bar{x}$ are normal random vectors provided that $\bar{x}$ is a normal random vector. Find $E(\bar{y})$ and $V(\bar{y})$. ...
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2answers
31 views

Central value of the partial exponential function [duplicate]

I need help calculating the central value of the partial exponential function : $$\lim_{n \to \infty} e^{-n} \sum^n_{k=0} \frac{n^k}{k!}$$ fd
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1answer
209 views

Is it possible to derive the CDF of $Z$?

Assume that $X_i$, $Y_k$, $i=0,\ldots,N$, $k=1,\ldots,K$ are non-negative independent non-identically distributed random variables. Let us define the random variable $Z$ as \begin{align} ...
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1answer
125 views

Conditional Expected Value of Product of Normal and Log Normal Distribution

Could someone please provide the answer and steps to solve this expression? \begin{eqnarray*} E\left[\left.\left(e^{X}Y+k\right)\right|\left.\left(e^{X}Y+k\right)>0\right]\right. \end{eqnarray*} ...
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2answers
369 views

Does variance do any good to gambling game makers?

People always like to evaluate the variance, but is there any way for variance to be interesting to the gambling game makers? In another word, what is a pratical gambling game that involving some ...
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5answers
1k views

Probability distribution for distances between randomly selected integers within an interval

Suppose I pick 'N' integers over an interval [A, B] without replacement. As a function of 'N' and the interval length, what distribution / average values should I expect for the distances between ...
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1answer
201 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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2answers
168 views

Are there any (pairs of) simple distributions that give rise to a power law ratio?

If I recall correctly, for $X$, $Y$ normally distributed, the ratio $X/Y$ is Cauchy-distributed. This is sort of like a power law, but isn't quite. So: Are there any simple distributions for two ...
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1answer
277 views

How can I prove this expression not to be a characteristic function

Let $\phi$ be a function of two real arguments defined as follows: $$ \phi\left(t_1, t_2\right) = \exp \left(-t_1^2-t_2^2 +i \frac{t_1}{3}\frac{ t_1^2-3 t_2^2 }{t_1^2+t_2^2} \right)$$ and whenever ...
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1answer
187 views

An interesting inequality about the cdf of the normal distribution

When approaching this other question I came out with the inequality: $$\frac{1}{4+x^2}e^{-x^2/2} \leq\Phi(x)\Phi(-x)\leq \frac{1}{4}e^{-x^2/2},\tag{1}$$ where $\Phi(x)$ is the cdf of the standard ...
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1answer
156 views

Characterizing a distribution by its projections

Consider the density $f(x,y)=\large\frac{1}{2\pi}\frac{1}{\sqrt{1-x^2-y^2}}$ on the unit disk centered at the origin. There is a particular characterization of this distribution: it is the unique ...
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3answers
753 views

Product of random variables

$X,Y$ and $Z$ are independent uniformly distributed on $[0,1]$ How is random variable $(XY)^Z$ distributed? I had an idea to logarithm this and use convolution integral for the sum, but I'm not sure ...
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0answers
177 views

Need advice: what should be my next step?

I am dealing with a quite algebraic question and I arrived at some good point. I had $2$ equations with $2$ unknowns and I was able to eleminate one of the variables. My final equation still seems ...
5
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2answers
3k views

Correlated Poisson Distribution

$X_1$ and $X_2$ are discrete stochastic variables. They can both be modeled by a Poisson process with arrival rates $\lambda_1$ and $\lambda_2$ respectively. $X_1$ and $X_2$ have a constant ...
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2answers
73 views

CDF of probablity distribution with replacement

I want to get every color of gumball in a gumball machine (where there are 16 types of gumballs, each with a 1/16 chance of obtaining a particular color [assume there are an infinite amount of ...
4
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1answer
107 views

Slowly varying function without limit at infinity

A function $f:\mathbb R \to \mathbb R$ is slowly varying at infinity if for any $t>0$ $$ \lim_{x\to +\infty}\frac{f(xt)}{f(x)}=1. $$ Is there a bounded function slowly varying at infinity whose ...
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1answer
488 views

Maximum order statistic for Binomial distribution

Let $X_i$, $1\le i\le t$, be $t$ independent random variables with Binomial distribution $B(n,\frac1t)$. I would like to find the distribution of $X_{Max}=\max_{i=1}^t(X_i)$ Note that this is the ...
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1answer
2k views

If $X$ is a Poisson distribution with mean $\lambda$ how is $X^2$ distributed?

If $X$ is a Poisson distribution with mean $\lambda$ how is $X^2$ distributed? Any explanation would be very appreciated.
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2answers
167 views

Verifying Exponential Family

Why is the following $$f(x|\theta) = \theta^{-1} \exp(1-(x/\theta)), \ \ 0 < \theta < x <\infty$$ not an exponential family? We know that $$f(x| \theta) = h(x)c(\theta) \exp(w(\theta)t(x))$$ ...
4
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3answers
663 views

Top 3 of 4 Dice Rolls

I'm trying to prove why the mean of the distribution of sums of the top 3 out of 4 fair 6 sided dice is rolls 12.25. Anybody who's rolled a D&D character knows the idea. $r_n = Rand([1,6])$ $x ...
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3answers
297 views

Probability distribution function

I am trying to develop a function that will allow me to input a random number between 0 and 1 and receive a value. The idea is that the function has a range (for example, 0-100) with a median value of ...
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3answers
2k views

calculate the Probability density function of the absolute difference of two random variable

If $X$ and $Y$ are two independent random variables with probability density functions $f$ and $g$, respectively, then the probability density of the difference $Y − X$ is given by the ...
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4answers
8k views

Convolution of two Gaussians is a Gaussian

I know that the product of two Gaussians is a Gaussian, and I know that the convolution of two Gaussians is also a Gaussian. I guess I was just wondering if there's a proof out there to show that the ...
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0answers
35 views

Distribution and convergence of the r.vs.: $X_n= \frac{ \lfloor nX \rfloor}{n}$

$X$ is an absolutely continous random variable, with a continous density function, and: $$X_n= \frac{ \lfloor nX \rfloor}{n}$$ What is the distribution of $X_n$, and what can we say about its ...
3
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1answer
94 views

Determine the distribution of $\int_0^t (W_s-\frac{s}{t}W_t) ds$, where $(W_s)_{s\geq 0}$ is a brownian motion

I have to find the distribution of $X_t:=\int_0^t (W_s-\frac{s}{t}W_t) ds$ where $(W_s)_{s\geq 0}$ is a brownian motion. I already showed the first integral $\int_0^t W_s ds$ is ...
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1answer
205 views

Convergence of marginal distribtution

Here I have a question which looks a little bit weird: $(q_n)_n$ is sequence of probability density functions of the couple $(x,y) \in \mathbb R^2$, $p_n$ is the marginal density of $q_n$, i.e. ...
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1answer
142 views

Exponentially distributed random variables

Given two exponentially distributed random variables $ X_1 $ and $X_2$ (assuming rates $\lambda_1$ and $\lambda_2$ respectively), determine the probability that one is smaller than the other. So ...
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1answer
607 views

Result and proof on the conditional expectation of the product of two random variables

My problem is the following: $X$ and $Y$ are two random variables and $\mathcal{F}$ is a $\sigma$-algrebra. Given that $X$ and $Y$ are independant, and that $X$ is independant of $\mathcal{F}$, can I ...
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3answers
1k views

X,Y are independent exponentially distributed then what is the distribution of X/(X+Y)

Been crushing my head with this exercise. I know how to get the distribution of a ratio of exponential variables and of the sum of them, but i can't piece everything together. The exercise goes as ...
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2answers
187 views

Mean of an increasing function over exponential distribution

I came across the following problem in my research I have two random variables $X, Y$ which are exponentially distributed and $Y$ has a higher mean than $X$. Then I have a function, say $f(z)$, ...
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3answers
342 views

$E[X]$ finite iff $\sum\limits_{n} P(X>an)$ converges

Show that: $$\sum\limits_{n \in N } P(X>an) < \infty\ \text{for some}\ a > 0 \Rightarrow E[X] < \infty \Rightarrow \sum\limits_{n \in N } P(X>an) < \infty\ \text{for every}\ a > ...
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0answers
162 views

Simplifying covariance matrices in distributions

In the multivariate Gaussian distribution, it is required that the covariance matrix be positive semidefinite. I have read that a positive semidefinite matrix $\Sigma$ can be written as $LL^{T}$. I ...
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3answers
627 views

Expected number of overlaps between intervals

Suppose $N$ intervals of length $\delta$ are positioned in $[0,1]$. The starting point $l_i$ of each interval is drawn from an uniform distribution, i.e., $l_i \in [0, 1-\delta]$, thus it will ...
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3answers
4k views

The mode of the Poisson Distribution

Lately, I am doing an investigation on Stirling's formula and its applications. So I thought I could use it to prove that the mode of the Poisson model is approximately equal to the mean. Of course, ...
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1answer
1k views

Four-parameter Beta distribution and Wikipedia

Sorry if it is not an appropriate place for such questions, but anyway can anybody please confirm that the formula for the density function of the four-parameter Beta distribution is correct in ...
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3answers
2k views

$3\sigma$ rule for multivariate normal distribution

I was wondering if the $3\sigma$ rule that holds for 1D normal distribution also holds for multivariate normal distribution?
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3answers
199 views

What is the distribution of $x'Cx$ when $x$ is a standard gaussian vector

When $x$ is an '$n$' dimensional standard Gaussian, we have $x'x \sim \chi^2$ with $n$ degrees of freedom. Now if I have a symmetric matrix $C$, what will be the distribution of $x'Cx$ ? ...
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0answers
234 views

Bayesian Estimation with Two Parameters

Warning After hours of trying, it has been proven (thanks, @leonbloy) that my attempt at a solution contained lots of mistakes. Maybe the correct answer is that there is no solution, but I don't ...