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.
9,021 questions
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Result of repeated applications of the binomial distribution?
What is the result of multiplying several (or perhaps an infinite number) of binomial distributions together?
To clarify, an example.
Suppose that a bunch of people are playing a game with k (to ...
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Conditional expectation of convolution product equals..
Let $X, Y$ be two $L^1$ random variables on the probablity space $(\Omega, \mathcal{F}, P)$. Let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-algebra.
Consider the conditional expectation ...
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A Local CLT with large variance
For n an even integer, $0 \leq i \leq$ ${n} \choose{j}$, $1 \leq j \leq n$ let $X_{i,j}$ be a
random variable taking values $\frac{n}{2}-j,0,j - \frac{n}{2}$ with equal probability. Let $S_{n}$ be ...
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A probabilistic inequality [closed]
Suppose $x_1,x_2,...,x_6$ are non-negative Independent and identically-distributed random variables, is it true that $P(x_1+x_2+x_3+x_4+x_5+x_6 \lt 3\delta) \lt 2P(x_1 \lt \delta)$ for any $\delta \...
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Parity, Balls and Boxes
Start with a distribution $\mu$ on [n], and drop m balls into these n+1 slots independently and according to the distribution &mu. That is, we have iid random variables x 1 through x m ...
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Difference Equations & Possible Limits
The answer to this may well be in some elementary textbook - a reference might be more useful than a short answer here.
If we look at the behaviour of a point in R n under matrix multiplication, we ...
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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?
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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,...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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 &...
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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 ...
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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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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 ...
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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 ≤...
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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.
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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 ...
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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(...
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