All Questions
9,497 questions
0
votes
2
answers
449
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,...
- 123
5
votes
5
answers
3k
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 $Corr(e_n,m_n)=p$. ...
- 109
5
votes
3
answers
3k
views
Formula for the nth convolution of a Laplace random variable
Let $x_1, x_2, ...$ be i.i.d. 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 $...
- 109
14
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 $...
- 2,302
4
votes
3
answers
791
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 ...
3
votes
2
answers
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 $\alpha$, $\beta$?
- 597
4
votes
4
answers
3k
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?
- 5,217
21
votes
6
answers
3k
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 $3$ ...
- 68.8k
0
votes
1
answer
485
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 ...
191
votes
34
answers
81k
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
1k
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 ...
7
votes
7
answers
2k
views
CLT for stationary sequences with infinite variance
There is a well-known central limit theorem for as a stationary sequences.
If $( X_n )_n$ is a stationary sequence and $E X_n=0$ then under suitable mixing conditions the sequence $S_n := n^{-1/2}\...
- 1,491
2
votes
2
answers
6k
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 ...
- 5,217
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 $\frac{X_1+\dots+X_n}{\sqrt{n}}$ ...
- 5,781
9
votes
3
answers
5k
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)...
- 1,491
-2
votes
2
answers
245
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
3
votes
3
answers
2k
views
Is the min function ever an unbiased estimator for the mean?
Given $n$ i.i.d. variables $X_1$ to $X_n$ with an unknown probability distribution, the sample average is an unbiased estimator for the mean of the distribution. Is there some non-trivial probability ...
- 597
2
votes
3
answers
826
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 ...
- 3,613
12
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?
- 139
15
votes
8
answers
3k
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 ...
- 24.7k
8
votes
3
answers
1k
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 ...
- 156k
17
votes
4
answers
762
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
751
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]} |...
- 1,491
6
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 ...
- 976
2
votes
2
answers
1k
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 ...
- 13k
26
votes
5
answers
10k
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 ...
- 14k
72
votes
9
answers
30k
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. ...
- 2,649
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 ...
- 5,935
9
votes
3
answers
354
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 ...
- 811
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 ...
- 14k
29
votes
5
answers
9k
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, ...
- 2,615
4
votes
2
answers
543
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 ...
- 41
4
votes
4
answers
1k
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||...
- 2,649
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 ...
- 61.9k
6
votes
3
answers
790
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 ...
- 811
11
votes
3
answers
5k
views
Strong law of large numbers for weakly dependent random variables
Let $X_i$ be a sequence of identically-distributed random variables with finite-range dependence (i.e. there exists $I$ such that if $|i-i'| \ge I$, then $X_i$ and $X_{i'}$ are independent), and a ...
- 8,502
91
votes
13
answers
146k
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 ...
- 14k
9
votes
1
answer
10k
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.
- 117
-1
votes
1
answer
338
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 ≤...
- 117
16
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 ...
- 2,150
36
votes
2
answers
13k
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 ...
- 811
11
votes
3
answers
4k
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?
$$erfc(x) > \frac{ x \exp(-x^...
- 5,217
9
votes
6
answers
3k
views
Primes are pseudorandom?
I've been reading the wonderful slides by Terry Tao and thought about this question.
Primes appear to be quite random, and the formal statement should be that there are some characteristics of primes ...
- 15.1k
9
votes
4
answers
850
views
easy(?) probability/diff eq. question
I've been wondering about this ever since I was a little kid and I used to ride in the back of the car and my mom would speed like hell towards a green light, only to slam on the brakes when she ...
- 6,069
10
votes
4
answers
966
views
What m minimizes E(|m-X|^3) for a random variable X?
Let X be a random variable. Then E(|m-X|^1) is minimized when (as a function of m) when m is the median of X, and E(|m-X|^2) is minimized when m is the mean of x.
A couple weeks ago in a technical ...
- 14k
49
votes
13
answers
24k
views
Why is it so cool to square numbers (in terms of finding the standard deviation)?
When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do
$$\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 ...
- 673
2
votes
2
answers
372
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(...
- 23