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

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47
votes
9answers
9k 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: ...
39
votes
4answers
52k 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
622 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
3answers
27k 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 ...
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 $x$. ...
23
votes
2answers
5k 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 ...
20
votes
3answers
12k 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
5answers
26k 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 ...
18
votes
2answers
405 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 ...
17
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 ...
17
votes
4answers
913 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 ...
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 .. ...
16
votes
1answer
7k 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 ...
16
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 ...
16
votes
1answer
274 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 ...
15
votes
2answers
317 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
661 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} ...
15
votes
2answers
324 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 ...
14
votes
3answers
7k 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 ...
14
votes
2answers
910 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 ...
14
votes
0answers
169 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 ...
13
votes
4answers
13k 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
1answer
2k 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 ) $ ...
13
votes
3answers
580 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 ...
13
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, ...
13
votes
1answer
279 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
2answers
9k 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
920 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
1answer
104 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 ...
12
votes
3answers
603 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
949 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 ...
11
votes
3answers
941 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
5answers
577 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 ...
11
votes
3answers
12k 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
3answers
802 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 ...
11
votes
1answer
13k 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 ...
11
votes
1answer
511 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 ...
11
votes
1answer
213 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
457 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 ...
10
votes
2answers
649 views

What is the new probability density function by generating a random number by taking the reciprocal of a uniformly random number between 0 and 1?

I have a random number generator which can generate a random number between $0$ and $1$. I attempt to generate a random number between 1 and infinity, by using that random number generator, but ...
10
votes
6answers
385 views

Die that never rolls the same number consecutively

Suppose we have a "magic" die $[1-6]$ that never rolls the same number consecutively. That means you will never find the same number repeated in a row. Now let's suppose that we roll this die $1000$ ...
10
votes
3answers
11k views

How does a geometric distribution converge to an exponential distribution?

I am trying to define an indexing $n$/$m$ and I am sending $m$ to infinity, but I get zero...not some relevant distribution. What is the technique or approach one must use here?!
10
votes
1answer
442 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
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 ...
10
votes
5answers
12k 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
12k views

How exactly are the beta and gamma distributions related?

According to Wikipedia, the Beta distribution is related to the gamma distribution by the following relation: $$\lim_{n\to\infty}n B(k, n) = \Gamma(k, 1)$$ Can you point me to a derivation of this ...
10
votes
2answers
449 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
405 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
596 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 ...