# 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

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votes

**1**answer

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### 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 ...

**0**

votes

**2**answers

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,...

**5**

votes

**5**answers

2k views

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

Suppose you have two simple Ar[1] 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 $Corr(e_n,m_n)=p$. ...

**4**

votes

**2**answers

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 ...

**12**

votes

**4**answers

2k views

### How long for a simple random walk to exceed $\sqrt{T}$?

Let $R_n$ be a simple random walk with $R_0 = 0$, and let $T$ be the smallest index such that $k\sqrt{T} < |R_T|$ for some positive $k$.
What is an expression for the probability distribution of $...

**4**

votes

**3**answers

536 views

### order statistics for components of a random unit vector

Suppose you sample uniformly from the unit vectors in R^n. What are the distributions of the order statistics of the magnitudes of the components of the sampled vectors? That is, for 1 <= i <= n ...

**1**

vote

**1**answer

2k views

### Expected value of a gamma-distributed random variable to the n-th power?

Is there a closed form for $E(Y^n)$, where $Y$ is a random variable with a gamma distribution with parameters $\alpha$, $\beta$?

**3**

votes

**4**answers

2k views

### Harmonic mean of random variables

The arithmetic mean of normal random variables is normal. The geometric mean of log-normal random variables is log-normal. But is there a common distribution family closed under taking harmonic means?

**20**

votes

**6**answers

2k views

### “The” random tree

One time I heard a talk about "the" random tree. This tree has one vertex for each natural number, and the edges are constructed probabilistically. Connect vertex $2$ to vertex $1$. Connect vertex $3$ ...

**0**

votes

**1**answer

363 views

### Estimating probability of set membership

I have a number of discrete finite sets, $A_0$ through $A_n$. I do not actually know their contents, but I know the size of each set and the size of the intersection between $A_0$ and each of the ...

**147**

votes

**33**answers

54k views

### What is convolution intuitively?

If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...

**4**

votes

**4**answers

879 views

### Deconvolution of gamma distributions

If the sum of two independent random variables is gamma distributed does this imply that the individual random variables are also gamma distributed. I suspect that the answer is no, but I do not know ...

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votes

**7**answers

1k views

### CLT for stationary sequences with infinte variance

There is a well-known central limit theorem for as a stationary sequences.
If $( X_n )_n$ is a sationary sequence and $E X_n=0$ then under suitable mixing conditions the sequence $S_n := n^{-1/2}\...

**2**

votes

**2**answers

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 ...

**9**

votes

**3**answers

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 $\frac{X_1+\dots+X_n}{\sqrt{n}}$ ...

**8**

votes

**2**answers

3k views

### Laplace transform and fractional moments.

Is there any "easy" way to calculate fractional moments from Laplace transform.
To be more specyfic let us consider the following example. Let $X$ be a positive random variable and
$L(\theta) := E \...

**-2**

votes

**2**answers

223 views

### Evaluate a fair game [closed]

I'm not a mathematician, so my question may be not so clear, sorry.
Let's say we toss up an ideal coin and win 1 dollar on heads and lose 1 dollar on tails. So, expected value is M = 1×0.5 &...

**1**

vote

**2**answers

525 views

### Is the min function ever an unbiased estimator for the mean?

Given n iid variables X1 to Xn with an unknown probability distribution, the sample average is an unbiased estimator for the mean of the distribution. Is there some non-trivial probability ...

**2**

votes

**3**answers

578 views

### When do binomial distributions occur?

A binomial distribution is the distribution of the number of successes of n independent, identical Bernoulli trials. What happens when the trials are dependent and the Bernoulli trials are not ...

**11**

votes

**5**answers

1k views

### What is hidden in Hidden Markov Models? [closed]

Why the word "hidden" present in hidden markov model? What exactly is hidden.
Whatever is hidden in HMM isn't it hidden in normal Markov Models?

**15**

votes

**8**answers

2k views

### How Does Random Noise Typically Look?

How does random noise in the digital world typically look?
Suppose you have a memory of n bits, and suppose that a "random noise" hits the memory in such a way that the probability of each bit being ...

**8**

votes

**3**answers

995 views

### Singularity of sparse random matrices

The following topic came up in conversation with my office-mate Lionel: Let $p$ be a fixed prime, $c$ a fixed positive real parameter and $n$ a large number. Consider a random $(0,1)$ matrix with ...

**17**

votes

**4**answers

723 views

### How many dimensions is it safe to get drunk in?

In Michael Lugo's blog post Variations on the drunken-bird theorem, and real-world sightings he wonders (without coming to a conclusion) what the maximum 'safe' number of dimensions to get drunk in ...

**2**

votes

**1**answer

455 views

### Suprema of stochastic processes

Let X be a continuous stochastic process. I know that (t>s)
P(|X(t) - X(s)|>δ) < |t-s|/δ
Is it possible to say anything (e.g. estimate the decay of the tail) about
Y=sup_{s \in [0,1]} |...

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votes

**3**answers

2k views

### question on sigma-fields

Let X,Y to be mappings from the sample space Ω to R and suppose Y is measurable with respect to σ(X), the smallest σ-field that makes X measurable.
Does it follow that there exists ...

**1**

vote

**1**answer

825 views

### Decoupling lemma for the Lambda(p) problem

I'm attempting to work through Bourgain's paper "Bounded orthogonal systems and the $\Lambda(p)$-set problem". There is a step in the proof of the decoupling lemma that I am stuck on, and thought ...

**14**

votes

**4**answers

6k views

### Is there a simple way to compute the number of ways to write a positive integer as the sum of three squares?

It's a standard theorem that the number of ways to write a positive integer N as the sum of two squares is given by four times the difference between its number of divisors which are congruent to 1 ...

**46**

votes

**8**answers

15k views

### When are probability distributions completely determined by their moments?

If two different probability distributions have identical moments, are they equal? I suspect not, but I would guess they are "mostly" equal, for example, on everything but a set of measure zero. ...

**6**

votes

**3**answers

4k views

### Rigorous definition, detection and test for trending vs. mean-reverting behaviour of stochastic processes

This is a question that has haunted me for some time. In the domain of time series you always talk about trends and mean reversion. But at least to me these concepts are either defined axiomaticly ...

**9**

votes

**3**answers

345 views

### Surfaces that are 'everywhere accessible' to a randomly positioned Newtonian particle with an arbitrary velocity vector

Consider an idealized classical particle confined to a two-dimensional surface that is frictionless. The particle's initial position on the surface is randomly selected, a nonzero velocity vector is ...

**19**

votes

**9**answers

3k views

### How can I generate random permutations of [n] with k cycles, where k is much larger than log n?

I've been thinking a lot lately about random permutations. It's well-known that the mean and variance of the number of cycles of a permutation chosen uniformly at random from Sn are both ...

**24**

votes

**5**answers

7k views

### Examples where Kolmogorov's zero-one law gives probability 0 or 1 but hard to determine which?

Inspired by this question, I was curious about a comment in this article:
In many situations, it can be easy to
apply Kolmogorov's zero-one law to
show that some event has probability 0
or 1, ...

**4**

votes

**2**answers

510 views

### maximizing function (stochastic calculus)

S is a price process which follows Geometric Brownian motion with no drift:
dS=S*vol*dW, vol=const., W is a Wiener process.
Define the following ratio: R=E[Max(f(S)-S(T),0)]/E[f(S)], where S(T) is ...

**4**

votes

**4**answers

952 views

### Distribution of 1-norm for Gaussian Unitary Ensemble

Suppose I uniformly sample matrices X from the Gaussian Unitary Ensemble (GUE) with variance \sigma^2. Consider the Ky-Fan d norm, i.e. the sum of the singular values, of X. Let's call this Z=||X||...

**12**

votes

**3**answers

1k views

### Expectation of the product of almost independent Gaussians

Let $X_ i$ be copies of the standard real Gaussian random variable $X$. Let $b$ be the expectation of $\log |X|$. Assume that the correlations $EX_ iX_ j$ are bounded by $\delta_ {|i-j|}$ in absolute ...

**6**

votes

**3**answers

688 views

### 'Focusing' the mass of the Probability Density Function for a Random Walk

Consider a random walk on a two-dimensional surface with circular reflecting boundary conditions (say, of radius 'R'). Here, for a fixed-size area, one finds a larger fraction of the probability ...

**10**

votes

**3**answers

4k views

### Strong Law of Large Numbers for weakly dependent random variables

Let Xi be a sequence of identically-distributed random variables with finite-range dependence (i.e. there exists I such that if |i-i'| ≥ I, then Xi and Xi' are independent), and a finite moment-...

**64**

votes

**12**answers

92k views

### If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this?

There is a standard problem in elementary probability that goes as follows. Consider a stick of length 1. Pick two points uniformly at random on the stick, and break the stick at those points. What ...

**4**

votes

**1**answer

7k views

### What is the difference between a homogeneous stochastic process and a stationary one?

Hello.
I am studying stochastic process.
here,
I don't know what is difference of
"the process is homogeneous"
and
"the process is stationary"
I feel confusing. It seems to similar to me.

**-2**

votes

**1**answer

320 views

### about Function of Random variables [closed]

Hello,
I am studying random variables.
Question is this:
if rv X & a function g is known, what is the pdf of random variable Y = g(x)?
in the textbook answer is explained as follows.
P[y ≤ Y ≤...

**-1**

votes

**1**answer

422 views

### multidimensional multinomial density [closed]

I have data set X = {x_1, x_2, \ldots, x_N}, each x_i
is a d-dimensional vector, where scalars are from some finite field
(In practice they are categories, represented by integers from 1...C).
If ...

**13**

votes

**6**answers

3k views

### analog of principle of inclusion-exclusion

When I teach elementary probability to my finite math students, a common error is to mix up the concepts of disjointness and independence. At some point I thought that it might be helpful to some ...

**29**

votes

**2**answers

7k views

### Mean minimum distance for N random points on a one-dimensional line

Let's say that I have a one-dimensional line of finite length 'L' that I populate with a set of 'N' random points. I was wondering if there was a simple/straightforward method (not involving long ...

**9**

votes

**3**answers

2k views

### erfc lower bound

I've seen the following lower bound for the complementary error function (erfc) but I haven't been able to prove it. Does anyone know how to establish the following?
$$erfc(x) > \frac{ x \exp(-x^...

**7**

votes

**6**answers

2k views

### Primes are pseudorandom?

I've been reading the wonderful slides by Terry Tao and thought about this question.
Primes appear to be quite random, and the formal statement should be that there are some characteristics of primes ...

**10**

votes

**4**answers

793 views

### easy(?) probability/diff eq. question

I've been wondering about this ever since I was a little kid and I used to ride in the back of the car and my mom would speed like hell towards a green light, only to slam on the brakes when she ...

**9**

votes

**4**answers

769 views

### What m minimizes E(|m-X|^3) for a random variable X?

Let X be a random variable. Then E(|m-X|^1) is minimized when (as a function of m) when m is the median of X, and E(|m-X|^2) is minimized when m is the mean of x.
A couple weeks ago in a technical ...

**35**

votes

**12**answers

17k views

### Why is it so cool to square numbers (in terms of finding the standard deviation)?

When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do
$$\sigma = \sqrt{ {1 \over 5-1} \left( (1-2.6)^2 + (2-2.6)^2 + (2-2.6)^2 + (3-2.6)^2 + (5 - 2.6)^2 \right) } \approx 1.52$$.
Why ...

**2**

votes

**2**answers

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(...