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

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44
votes
9answers
8k 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: ...
36
votes
4answers
44k views

Probability density function vs. probability mass function

I've a 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 ...
29
votes
1answer
597 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 ...
24
votes
2answers
2k 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 $x$. ...
21
votes
3answers
24k 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 ...
20
votes
1answer
3k views

How was the normal distribution derived?

Abraham de Moivre, when he came up with this formula, had to assure that the points of inflection were exactly one standard deviation away from the center, and so that it was bell-shaped, as well as ...
19
votes
3answers
10k 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 ? ...
18
votes
2answers
401 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 ...
16
votes
5answers
22k views

Poisson Distribution of sum of two random independent variables $X$, $Y$

$X \sim \mathcal{P}( \lambda) $ and $Y \sim \mathcal{P}( \mu)$ meaning that $X$ and $Y$ are Poisson distributions. What is the probability distribution law of $X + Y$. I know it is $X+Y \sim ...
16
votes
3answers
7k views

Expectation of the maximum of i.i.d. 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 .. ...
15
votes
3answers
4k views

How to deduce the CDF of $W=I^2R$ from the PDFs of $I$ and $R$ independent

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 ...
15
votes
2answers
3k views

There are 10 men, 10 women, and 10 rooms. Each person randomly goes into a room.

What is the expected number of rooms with at least one man and woman? Our prof. gave us the following solution however, I'm confused about the probability portion of the answer (especially the ...
15
votes
2answers
310 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 ...
15
votes
1answer
269 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
1answer
6k 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 ...
14
votes
3answers
859 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
4answers
10k 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 ...
13
votes
2answers
851 views

Existence of independent and identically distributed random variables.

I often see the sentence "let $X_1, X_2, \ldots$ be a sequence of i.i.d. random variables with a certain distribution". But given a random variable $X$ on a probability space $\Omega$, how do I know ...
13
votes
1answer
278 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 ...
13
votes
2answers
263 views

Probability of picking an odd number from the set of naturals?

I know there's no uniform distribution for a countably infinite set, but I'm wondering if there's still a way to determine the probability of picking from a subset of a countably infinite set. For ...
13
votes
0answers
506 views

Integrate this monster

Can you please help me? I've been trying for some time now to integrate this: $$\int_0^\infty g^{-(a+1)} \; \exp\left\{-\left(\frac{b}{g} + \frac{1}{2} \sum_{i=1}^{n} ...
12
votes
2answers
8k views

Proof of upper-tail inequality for standard normal distribution

$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$, $$ \mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>. $$
12
votes
3answers
6k views

Integral of Brownian motion is Gaussian?

Let $(W_t)$ be a standard Brownian motion, so that $W_t \sim N(0,t)$. I'm trying to show that the random variable defined by $Z_t = \int_0^t W_s \ ds$ is a Gaussian random variable, but have not ...
12
votes
3answers
550 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 ...
12
votes
3answers
830 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 ...
12
votes
3answers
595 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 ...
12
votes
1answer
3k views

The partial expectation $\mathbb{E}(X;_{X>K})$ for an alpha-stable distributed random variable

The partial expectation $\mathbb{E}(X;_{X>K})$ for an alpha-stable distributed random variable: By playing with convolutions of Characteristic Functions of alpha-Stable distributions $S(\alpha, ...
11
votes
3answers
876 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
1k views

Formal definition of conditional probability

It would be extremely helpful if anyone gives me the formal definition of conditional probability and expectation in the following setting, given probability space $ (\Omega, \mathscr{A}, \mu ) $ ...
11
votes
3answers
11k 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 ...
11
votes
1answer
96 views

Distribution of $\sum\limits_{i=1}^{N}X_{i}$ conditionally on $\sum\limits_{i=1}^{N}X_{i}^{2}$ for i.i.d. standard normal $X_i$s

Assume that the random variables $X_{i}$ are i.i.d $\mathcal{N}\left(0,1\right)$, then: $$S_N=\sum_{i=1}^{N}X_{i}\sim\mathcal{N}\left(0,N\right)\qquad\qquad ...
11
votes
1answer
203 views

About the “Cantor volume” of the $n$-dimensional unit ball

A simple derivation for the Lebesgue measure of the euclidean unit ball in $\mathbb{R}^n$ follows from computing $$ \int_{\mathbb{R}^n}e^{-\|x\|^2}\,dx $$ in two different ways. See, for instance, ...
11
votes
2answers
370 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 ...
11
votes
1answer
896 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
5answers
570 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 ...
10
votes
1answer
414 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?
10
votes
5answers
10k views

What is the use of moments in statistics

Can any one give an "simple" explaination about what is the use of moments in statistics.Why we need moments? what we can learn from it? if possible please use less equations. Advance thanks for your ...
10
votes
3answers
790 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
419 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
2answers
2k views

Biased Random Walk and PDF of Time of First Return

I have a random walk process where each step the probability of $+1$ is $p$ and $-1$ is $q$, with $p+q=1$. $p$ may not equal $q$. The walker starts at zero. I want to know the probability that the ...
10
votes
1answer
384 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
1answer
12k 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 ...
10
votes
4answers
590 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
317 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 ...
10
votes
1answer
487 views

Math Intuition and Natural Motivation Behind t-Student Distribution

I am trying to understand with basic mathematical background how the $t$-Student distribution is a "natural" $pdf$ to define. So I hope that this not too-general a question, but given that the ...
10
votes
3answers
194 views

Distribution and second order differential inequality

I would like to solve the following $2^{\mathrm{nd}}$ order differential inequality $$ \theta_F'(x) = \frac{2F'(x)^2 - F(x)F''(x) + F''(x)}{F'(x)^2} < 0 $$ for some subinterval $I \subset ...
10
votes
1answer
814 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 ...
10
votes
0answers
120 views

Integrating a matrix function involving a determinant and exponential trace

I am trying to find the normalizing constant for a probability distribution and ran into a difficult integral. When $X$ is an $p \times k$ matrix, $a>0,$ and $g>0,$ how can I compute $$\int ...
9
votes
5answers
3k 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 ...
9
votes
3answers
4k 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 ...