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

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38
votes
8answers
3k 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: ...
22
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 $x$. ...
17
votes
3answers
14k 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
360 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
1answer
216 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
2answers
275 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
572 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
245 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
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 .. ...
12
votes
2answers
9k 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 ...
12
votes
3answers
540 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
538 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
568 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
651 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
1answer
2k 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
3answers
3k 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 ? ...
10
votes
1answer
219 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
2answers
247 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
3answers
2k 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 ...
9
votes
2answers
215 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
3answers
295 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 ...
9
votes
4answers
491 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 ...
9
votes
1answer
189 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
691 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
6answers
530 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
353 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
322 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
1answer
140 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
881 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
1answer
265 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
219 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
287 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 ...
7
votes
1answer
936 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
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
488 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
180 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
166 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 ...
7
votes
1answer
164 views

If $X,Y,Z$ are iid unif[0,1], then $(XY)^Z \sim \text{unif}[0,1]$.

Here's a mind-blowing fact (to me at least) that is perhaps not so well-known: If $X, Y, Z$ are iid uniformly distributed in $[0,1]$, then $W = (XY)^Z$ is also uniformly distributed in $[0,1]$. If ...
7
votes
0answers
99 views

How well can the maximum of a Gaussian process be approximated by a finite-dimensional Gaussian variable?

Consider a compact set $K$ in $\mathbb{R}^p$, and let $W$ be a mean zero continuous Gaussian process on $K$, meaning that $W$ takes its values in the space of continuous functions from $K$ to ...
7
votes
1answer
497 views

median of the F-distribution

Is the median of the F-distribution with m and n degrees of freedom decreasing in n, for any m? From experiments it looks like it might be, but I have been unable to prove it.
6
votes
4answers
1k 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 ...
6
votes
2answers
745 views

Uniform distribution with probability density function. Find the value of $k$.

For a random sample $X_1,X_2,...X_n$ from a uniform $[0,\Theta]$ distribution, with probability density function $$f(x;\Theta) = \left\{ \begin{array} \ \frac{1}{\Theta} & 0\le x \le\Theta,\\ 0 ...
6
votes
2answers
3k views

Proof of upper-tail inequality for standard normal distribution

$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$, $$ \mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>. $$
6
votes
3answers
1k views

Quantile function properties

I am confused by "Inverse distribution function (quantile function)" section of the wikipedia page on CDFs . It says that $$F^{-1}(F(x)) \leq x\text{ and }F(F^{-1}(y)) \geq y$$ However, I ...
6
votes
3answers
7k views

Sum of independent Binomial random variables with different probabilities?

suppose I have independent random variables $X_i$ which are distributed binomially via $$X_i \sim \mathrm{Bin}(n_i, p_i)$$. Are there relatively simple formulae or at least bounds for the ...
6
votes
3answers
557 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 ...
6
votes
2answers
262 views

Beta Function — finding a lower bound based on parameters

I would like to show that $$ 1-\frac{1}{c}Beta\left(c+1,\frac{1}{c}\right) \geq \frac{1}{c+1}.$$ for all $c \geq 2$. I have plotted it out for $c$ up through 200, and it seems to hold. Does anyone ...
6
votes
1answer
1k views

Distribution of the digits of Pi

Can anything be stated about the distribution of the digits of Pi, i.e., if I were to sample n digits of Pi, can anything be said about the probability to observe certain digits, or is there any ...
6
votes
2answers
141 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 ...