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Results tagged with pr.probability
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user 9025
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
3
votes
Clusters of uniformly distributed random points
This 1965 paper of Naus contains formulas for the exact probabilities. Turning them into the asymptotic results you want might take a little effort but since Naus' paper is cited in at least 255 plac …
- 37k
3
votes
Conditional geometric distributions
Define $g(\gamma) = \sum_{j\in J} \gamma^j$. The condition $E(X^2)\sim A\mu^2$ as $\mu\to\infty$ seems to be equivalent to
$$ \frac{g(\gamma) g''(\gamma)}{(g'(\gamma))^2} \to A $$
as $\gamma\to 1$ fr …
- 37k
2
votes
Accepted
Large deviations for bernoulli sums
There are quite a lot of approximations more accurate than the pure normal distribution, with a large literature difficult to sort out. You can see several in this paper (Adv. Appl. Prob., 21 (1989) 4 …
- 37k
2
votes
many expected streaks imply high probability for a streak
The broken logic in the book can bite in many places. Here is a simple example. If we make a random graph with $n$ vertices and edge probability $3/n$, the expected number of hamiltonian cycles goes …
- 37k
2
votes
Accepted
Equal probability of having even/odd number of ones in many Bernoulli trials with different ...
The probability that an even number of trials succeed is exactly
$$ \frac12\biggl( 1 + \prod_{i=1}^n (1-2p_i)\Biggr).$$
This is a standard elementary application of probability generating functions.
- 37k
1
vote
Accepted
Tail Conditional Expectation of a binomial random variable
CLT (sufficiently powerful version such as Berry-Esseen inequality) says that $Pr(X\ge c)\to\frac12$, so any event that has tiny probability for $X$ also has tiny probability for $X_{\ge c}$. So $E(X …
- 37k
3
votes
2d moment of chebyshev
$E[X^{2d}]/(2d)!$ is the coefficient of $x^{2d}$ in the Taylor expansion of $\cosh(x)^n$,
since $\cosh(x)$ is the exponential moment generating function of each $X_i$.
I don't know if there is a clos …
- 37k
2
votes
Limits of binomial distribution
The total variation distance between $B(n,p)$ and Poisson($np$) goes to 0 as $n\to\infty$ whenever $p\to 0$. See http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2086772 for e …
- 37k
1
vote
Local limit theorem for Bernoulli sums
The distribution of $X$ is called the Poisson-Binomial distribution and searching on that phrase will find a large amount of literature on it.
- 37k
5
votes
Accepted
Does independence of the sequence $f(A_i, B)$ imply the sequence is independent of $B$?
It is true.
Let $\chi_Z(x,y)$ be the indicator function of the set $\lbrace (x,y) \,|\, f(x,y)\in Z\rbrace$ for some set $Z$.
The condition that the events $f(A_1,B)\in Z$ and $f(A_2,B)\in Z$ are in …
- 37k
6
votes
find the probability that an $n \times n$ determinant formed by taking the numbers $1, 2, \l...
(Partial answer)
In GF(2), row and column permutations preserve the determinant. The equivalence classes under those operations are non-isomorphic bipartite graphs with $n$ vertices on each side and $ …
- 37k
5
votes
Expected distance between points drawn from different distributions
Looking at the discrete nonnegative case, using your notation,
$$\sum_{i,j} (p_i-q_i)(p_j-q_j)|i-j|
= -2 \sum_{i=0}^\infty \left( \sum_{j=0}^i p_j - \sum_{j=0}^i q_j\right)^2
\le 0,$$
where there are …
- 37k
9
votes
Accepted
Expected value of logarithm of a binomial random variable
Just expand $ln(X+\alpha)$ as a Taylor series about $X=np$ and then do the binomial sum term by term. Use tail bounds on the binomial distribution to show that the error terms are meaningful. Without …
- 37k
14
votes
A dice probability question
Interestingly, the probability that $t$ occurs for a standard die $n=6$ as $t\to\infty$ is $\frac27$, with exponential convergence towards that value.
This is a standard generating function problem. T …
- 37k
2
votes
Accepted
Max Absolute Difference of Expectations under Change of Measure
By some continuity argument, we can assume that the weights $w_i$ are distinct.
Take any three indexes, say 1,2,3.
Increase $p_1$ by $\epsilon(w_3-w_2)$, $p_2$ by $\epsilon(w_1-w_3)$ and $p_3$ by
$\e …
- 37k