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

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38
votes
9answers
4k views

What do $\pi$ and $e$ stand for in the normal distribution formula?

I'm a very beginner in mathematics and there is one thing I've been wondering recently. The formula for the normal distribution is: ...
23
votes
2answers
1k views

Is there a uniform distribution over the real line?

For every interval $[a,b]$, there exists a uniform probability density over this interval, which is the constant function $f(x)=\frac{1}{|a-b|}$ for $a < x < b$, and $f(x)=0$ for all other $x$. ...
23
votes
1answer
430 views

Zombie outbreak on a $k$-regular graph

Suppose we have a zombie outbreak on a connected $k$-regular graph of order $n$. There are $n_0$ initially infected zombie nodes, and each turn, each zombie infects its neighbors with probability ...
20
votes
3answers
18k views

Probability density function vs. probability mass function

I've an confession to make. I've been using pdf's and pmf's without actually knowing what they are. The idea that I've been having so long is that density = area under the curve but if I look at it ...
17
votes
2answers
379 views

Angular distribution of lines passing through two squares.

Let's say I've got two squares with side length $d$ that are held parallel at a distance $m$ apart. Suppose that particles are randomly falling from above in such a way that the polar angle ...
15
votes
2answers
12k views

How can a probability density be greater than one and integrate to one

Wikipedia says: The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one. and it also says. Unlike a probability, a probability ...
15
votes
1answer
233 views

What is the distribution of this random series?

Let $\xi_n$ be iid and uniformly distributed on the three numbers $\{-1,0,1\}$. Set $$X = \sum_{n=1}^\infty \frac{\xi_n}{2^n}.$$ It is clear that the sum converges (surely) and the limit has $-1 \le ...
14
votes
3answers
4k views

Expectation of the maximum of IID geometric random variables

Given $n$ independent geometric random variables $X_n$, each with probability parameter $p$ (and thus expectation $E\left(X_n\right) = \frac{1}{p}$), what is $$E_n = E\left(\max_{i \in 1 .. ...
14
votes
2answers
290 views

How does one prove $\int_0^\infty \prod_{k=1}^\infty \operatorname{\rm sinc}\left( \frac{t}{2^{k+1}} \right) \mathrm{d} t = 2 \pi$

Looking into the distribution of a Fabius random variable: $$ X := \sum_{k=1}^\infty 2^{-k} u_k $$ where $u_k$ are i.i.d. uniform variables on a unit interval, I encountered the following ...
13
votes
3answers
636 views

very elementary proof of Maxwell's theorem

Maxwell's theorem (after James Clerk Maxwell) says that if a function $f(x_1,\ldots,x_n)$ of $n$ real variables is a product $f_1(x_1)\cdots f_n(x_n)$ and is rotation-invariant in the sense that the ...
13
votes
1answer
258 views

Help with a Bollobás proof - Switching between random graph models

I'm trying to make my way through Bollobás' book 'Models of Random Graphs', and unfortunately I've come entirely unstuck on one of his typical 2-line "and of course, this is entirely trivial"-style ...
12
votes
3answers
5k views

Expectation of the min of two independent random variables?

How do you compute the minimum of two independent random variables in the general case ? In the particular case there would be two uniforms variables with difference support, how should one proceed ? ...
12
votes
3answers
556 views

Formula for picking time closest to (but after) target

Let's say you have an arbitrary length of time. You are playing a game in which you want to push a button during this time span after a light comes on. If you do so, you win ($+1$), if not, you lose ...
11
votes
3answers
3k views

Given the pdf of independent RVs $I$ and $R$, how to find cdf of $W =I^2R$?

Given pdf of $I$ and $R$ (both $I$ and $R$ are independent RV's), how to find cdf of $W =I^2R$? Where, $$ \begin{align} f_I(i)&=6i(1-i), &0 \leq i \leq 1 \\ f_R(r)&=2r, &0 \leq ...
11
votes
3answers
585 views

Random point uniform on a sphere

If $X=(x,y,z)$ is a random point uniform on the unit sphere in $\mathbb{R}^3$, Are the coordinates $x$, $y$, $z$ uniform in interval $(-1,1)$?
11
votes
1answer
631 views

Probability density function of the integral of a continuous stochastic process

I am interested in whether there is a general method to calculate the pdf of the integral of a stochastic process that is continuous in time. My specific example: I am studying a stochastic given ...
10
votes
3answers
691 views

A very challenging probability question

In a certain 2-player game, the winner is determined by rolling a single 6-sided die in turn, until a 6 is shown, at which point the game ends immediately. Now, suppose that k dice are now rolled ...
10
votes
2answers
246 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 ...
10
votes
3answers
344 views

Are polynomials dense in Gaussian Sobolev space?

Let $\mu$ be standard Gaussian measure on $\mathbb{R}^n$, i.e. $d\mu = (2\pi)^{-n/2} e^{-|x|^2/2} dx$, and define the Gaussian Sobolev space $H^1(\mu)$ to be the completion of ...
10
votes
1answer
3k views

Characteristic function of a standard normal random variable

The characteristic function of a random variable X is given by $$\Phi_X(\omega) = \mathbb{E}e^{j\omega X}=\int_{-\infty}^\infty e^{j\omega x}f_X(x) dx.$$ One can easily capture the similarity between ...
10
votes
1answer
254 views

Distribution for random harmonic series

Consider random variable $X$ formed by the following infinite series: $X = \pm 1 \pm \frac{1}{2} \pm \frac{1}{3} \pm ... \frac{1}{n} ...$, where $+$ or $-$ sign for every summand is chosen ...
10
votes
4answers
533 views

Probability of global epidemic

Consider $\mathbb{Z}^2$ as a graph, where each node has four neighbours. 4 signals are emitted from $(0,0)$ in each of four directions (1 per direction) . A node that receives one signal (or more) at ...
10
votes
2answers
264 views

Problem about partial sum of exponential random variable

Let $X_1, X_2, \dots,X_n, X_{n+1}$ be independent random variable of exponential distribution, and the mean is 1. Let $S_i = X_1 + \dots + X_i$ I want to know ...
9
votes
4answers
2k views

Expected Value of a Binomial distribution?

If $\mathrm P(X=k)=\binom nkp^k(1-p)^{n-k}$ for a binomial distribution, then from the definition of the expected value $$\mathrm E(X) = \sum^n_{k=0}k\mathrm P(X=k)=\sum^n_{k=0}k\binom ...
9
votes
1answer
501 views

Is There a Continuous Analogue of the Hypergeometric Distribution?

As the title states, is there a continuous analogue of a Hypergeometric distribution? If $ X \sim H(m,n,N)$ is a common Hypergeometric distribution, where $N$ is the population size, $n$ is the ...
9
votes
1answer
195 views

$P(x,n) = \frac{x(n-1)!}{n^{x}(n-x)! }$ — What is the name of this probability distribution?

$$ P(x,n) = \frac{x(n-1)!}{n^{x}(n-x)! } $$ I'm having a really tough time describing what this distribution does, but it's simple in code. So if you know code, then read on: ...
9
votes
5answers
802 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 ...
8
votes
5answers
544 views

Successful approaches to the modelization of ''randomness''

If you pick a number $x$ randomly from $[0,100]$, we would naturally say that the probability of $x>50$ is $1/2$, right? This is because we assumed that randomly meant that the experiment was to ...
8
votes
1answer
371 views

M.SE reputation distribution

What distribution does the reputation points per user follow on math.SE (or on entire stackexchange)? Is there a mathematical explanation/model of it?
8
votes
2answers
353 views

What's the probability that three points determine an acute triangle?

Three distinct points are chosen at random from the unit square. The goal is to find the probability that they form an acute triangle. I started working on this because I want to know how to approach ...
8
votes
3answers
312 views

Probability distribution and their related distributions

I am taking a probability course right now and have encountered a lot of interesting distributions and their related distributions. For example, if we have two independent Poisson distribution ...
8
votes
3answers
214 views

distribution of $X^2 + Y^2$

Suppose $X$ and $Y$ are independent uniform distributions between $(0,1)$. What is the distribution of $X^2 + Y^2$? I derived that the pdf of $X^2$ is $\frac{1}{2\sqrt{x}}$ for $0\leq x \leq 1$. How ...
8
votes
2answers
379 views

Contradictory recursion

I encountered this problem which I find very contradictory to my intuition. Could anyone enlighten me why my recursive formulation is wrong? "Roll a fair dice until the game stops. The game stops ...
8
votes
1answer
9k views

comparing distribution of two data sets

I need to compare the distribution (unknown) of a set of data to the distribution of another one (unknown). In particular, I want to check for equality of the two distributions. What are some ...
8
votes
1answer
178 views

What is the intuition behind the generalized confidence interval?

What is the intuition behind the generalized confidence interval? My best description on GCI that it is the way to derive a formula to calcuate the area of the center region in a asymetry distribution ...
8
votes
3answers
1k views

“Go-first” dice for $N$ players

I'm interested in sets of dice that can be used to determine who "goes first" (hence the name) in an $N$-player game; more generally, I want to determine a complete ordering of the players with a ...
8
votes
2answers
140 views

Lies, damned lies, and statistics

A story currently in the U.S. news is that an organization has (in)conveniently had several specific hard disk drives fail within the same short period of time. The question is what is the likelihood ...
8
votes
2answers
1k views

Double exponential distribution is not an exponential family

Define a one-parameter exponential family as a family of densities of the form $$f_\theta(x)=\exp(\eta(\theta)T(x) + \xi(\theta))h(x)$$ where $T(x)$ and $h(x)$ are Borel functions, ...
8
votes
1answer
293 views

Repeatedly Toss Balls into Bins

$n$ balls are randomly tossed into $m$ bins, each bin can hold $k$ balls. If a ball is tossed into a full bin (already has $k$ balls in it), it can be tossed repeatedly and randomly into the $m$ bins ...
8
votes
1answer
308 views

Volume of the intersection of ellipsoids

How do I compute the volume of the intersection of two $n$-dimensional ellipsoids? Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid ...
7
votes
5answers
2k views

Conditional expectation of $\max(X,Y)$ and $\min(X,Y)$ when $X,Y$ are iid and exponentially distributed

I am trying to compute the conditional expectation $$E[\max(X,Y) | \min(X,Y)]$$ where $X$ and $Y$ are two iid random variables with $X,Y \sim \exp(1)$. I already calculated the densities of ...
7
votes
3answers
1k views

covariance of Gaussian Mixtures

I have a Gaussian mixture model, given by: $$ x \sim \sum_{i = 1}^M \alpha_i N(\mu_i, C_i) $$ Is there a way I can compute the overall covariance matrix if $x$? I would like to say "$x$ has a ...
7
votes
1answer
1k views

Limit using Poisson distribution [duplicate]

Show using the Poisson distribution that $$\lim_{n \to +\infty} e^{-n} \sum_{k=1}^{n}\frac{n^k}{k!} = \frac {1}{2}$$
7
votes
2answers
147 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
2k views

Minimum variance unbiased estimator for scale parameter of a certain gamma distribution

Let $X_1, X_2, ..., X_n$ be a random sample from a distribution with p.d.f., $$f(x;\theta)=\theta^2xe^{-x\theta} ; 0<x<\infty, \theta>0$$ Obtain minimum variance unbiased estimator of ...
7
votes
4answers
593 views

Distribution of maximum of partial sums of independent random variables

I have been looking all over the net to find a way to work out a probability distribution of a maximum of partial sums of independent random variables, but to no avail. So I have decided to try to ...
7
votes
1answer
263 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 ...
7
votes
1answer
49 views

Random walk on cubic lattice

Suppose at every point of the cubic grid in n dimensions is a particle, and at every timestep every particle moves at random to one of its 2n neighbours. As time goes to infinity, what is the ...
7
votes
1answer
181 views

Distribution of the maximum of noisy random variables

Let $X_1, \ldots, X_N$ be $N$ hidden random iid variables, all with the same standard distribution, let's say uniform $\mathcal{U}(0, 1)$ or Gaussians $\mathcal{N}(0, 1)$ (probably easiest). I observe ...
7
votes
0answers
248 views

How do you compute numerically the Earth mover's distance (EMD)?

I was trying to compute numerically (write a program) that calculated the EMD for two probability distribution $p_X$ and $q_X$. However, I had a hard time finding an outline of how to exactly compute ...