# Questions tagged [pr.probability]

Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

5,499 questions
Filter by
Sorted by
Tagged with
1answer
483 views

### Which iid variables give a normal variable, when multiplied?

Hello, I hope you'll find my riddle interesting. Z = XY Z ~ N(0, 1) X, Y are iid random variables (independent, identically distributed). We assume X and Y are symmetric. What is the distribution of ...
2answers
432 views

### Chances of streaks in small bit-streams

Let's say a series of 10 bits is output randomly. Now lets do that 256 times. I'd like to find out what the expected number of streaks of 1s or 0s are for each of the possible sizes 1-10. For example,...
5answers
2k views

### Computing correlation between time series with missing data.

Suppose you have two simple Ar series of the form $y_n=y_{n-1}+e_n$ and $x_n=x_{n-1}+m_n$, where $e_n$ and $m_n$ are normal white noise processes with no auto-correlation and $Corr(e_n,m_n)=p$. ...
2answers
2k views

### Formula for the nth convolution of a laplace random variable

Let x_1, x_2, ... be iid draws from a laplace distribution with scale parameter b. Is there a relatively nice closed form for x_1+x_2+...x_n? I've seen a derivation floating around for when b=1, but I ...
4answers
2k views

2answers
5k views

### Examples of random variables

I'm looking for a list of examples of random variables to use in teaching a measure-theoretic probability course. For example, the Rademacher functions are an explicit construction of independent ...
3answers
1k views

### When does a pointwise CLT hold?

Let $X$ be a random variable with mean $0$ and variance $1$, and let $X_1, X_2, X_3, \dots$ be iid copies of $X$. Under what conditions can we say that the density of $\frac{X_1+\dots+X_n}{\sqrt{n}}$ ...
2answers
3k views

2answers
339 views

### Limit of sequence involving gamma functions

Let G be the gamma function, and b be a constant in (-2,inf). Let H(n, i) = G(i+1+b) * G(n-i+1+b) / [G(i+1) * G(n-i+1)] for integers n > i > 0. Let S(n) = \sum_{i=1}^{i=n-1} H(n, i). Let x_ n = H(...