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

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0
votes
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 ...
0
votes
1answer
255 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 $3\sigma$ ...
0
votes
2answers
149 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.
0
votes
1answer
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 ...
0
votes
1answer
140 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,& ...
0
votes
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 ...
0
votes
2answers
774 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, ...
0
votes
1answer
132 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 $V(\bar{y})$. ...
0
votes
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
0
votes
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} ...
-1
votes
1answer
109 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*} ...
10
votes
2answers
359 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 ...
9
votes
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 ...
7
votes
1answer
191 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 ...
7
votes
2answers
167 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 ...
7
votes
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 ...
6
votes
1answer
182 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 ...
6
votes
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 ...
6
votes
3answers
749 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 ...
5
votes
1answer
54 views

Multivariate normal density function of function of random variable

Let $X_1,\dots,X_n$ be i.i.d random variables and $g$ be a symmetric function such that $$g(X_i,X_j)\sim N(\mu,\sigma^2)$$ for all $1\le i<j\le n$. I wish to know the density function of the joint ...
5
votes
0answers
175 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
votes
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 ...
4
votes
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
votes
1answer
101 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 ...
4
votes
1answer
478 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 ...
4
votes
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.
4
votes
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
votes
3answers
632 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 ...
4
votes
3answers
294 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 ...
4
votes
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 ...
4
votes
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 ...
3
votes
0answers
34 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
votes
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 ...
3
votes
1answer
203 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. ...
3
votes
1answer
140 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 ...
3
votes
1answer
591 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 ...
3
votes
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 ...
3
votes
2answers
183 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 $f(z)$, ...
3
votes
3answers
335 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 > ...
3
votes
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 ...
3
votes
3answers
618 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 ...
3
votes
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, ...
3
votes
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 ...
3
votes
3answers
198 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 $x'Cx$ ? ...
3
votes
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 ...
3
votes
1answer
131 views

The limit of the expectation of the top half+1 order stats of $n$ draws of $X$ as $n\to\infty$?

Can anyone help me compute the limit of the average of the top half +1 of order marginal order distribution of $n$ draws from $X$, as $i\to\infty$? Specifically, the limit as $i\to\infty$ of ...
3
votes
2answers
2k views

CDF of a ratio of exponential variables

Let $X$ and $Y$ be independent exponential variables with rates $\alpha$ and $\beta$, respectively. Find the CDF of $X/Y$. I tried out the problem, and wanted to check to see if my answer of: ...
3
votes
2answers
877 views

Random Exponential-like Distribution

Note: Not good at math and my terminology may be very wrong. I have a uniform random number generator that outputs a number between [0,1]. I'd like a function that returns a random number between 0 ...
2
votes
1answer
19 views

Ratio between $k$th highest number among $n$ and $n+1$ samples

Let $n\geq k$ be fixed positive integers, and let $X$ be a distribution on $[0,1]$ that is not the constant $0$ distribution. Let $E_n$ denote the expected value of the $k$th highest value among $n$ ...
2
votes
2answers
99 views

Random sums of iid Uniform random variables

Let $\{X_r : r\ge 1\}$ be independently and uniformly distributed on $[0,1]$. Let $0<x<1$ and define $$N=\min\{n\ge 1 : X_1 + X_2 +\ldots+X_n> x\}$$ Show that $$P(N>n) = ...