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Results tagged with pr.probability
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user 17114
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
4
votes
Limits of binomial distribution
I believe with a fairly standard scaling and shifting, we can recover a normal limit for any $0 < \alpha < 1$ by a standard application of the Lindeberg–Feller theorem.
Define the triangular array of …
- 1,136
9
votes
probability of IID sum being positive
No, this does not hold without the finite second-moment assumption, in general.
Consider the Levy $\alpha$-stable distributions, which will yield a whole family of examples. Using the parametrization …
- 1,136
16
votes
What is the expected value of an N-dim vector of uniform randoms that sum to 1 which have be...
Liviu has given an excellent answer with a nice geometric flavor to it. This answer is meant to serve as a complement with a slightly more probabilistic bent. In the process, some of the computations …
- 1,136
13
votes
Tail bound for Poisson random variable
A classical inequality of Teicher (1955) asserts
Proposition (Teicher). Let $X \sim \mathrm{Pois}(\lambda)$. Then, $\mathbb P(X \leq [\lambda]) > e^{-1}$.
A modification of his argument will all …
- 1,136
4
votes
Absolute moments of symmetrical distributions
This response attempts to address your second question.
I believe the following (a modification of a classical counterexample) is a counterexample, but it would be helpful to have it checked by other …
- 1,136
8
votes
A trick or a general technique? (Probabilistic Method)
Perhaps you're looking for something deeper, but does this not simply follow by monotonicity of the integral and the (almost sure) positivity of $Q$?
Namely,
$$
\mathbb E P = \mathbb E \frac{P}{Q} Q …
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7
votes
Accepted
Capped binomial random variables
The answer to your question is positive and, for example, follows immediately from Corollary 4 of
C. A. León and F. Perron (2003), Extremal properties of sums of Bernoulli random variables, Statis …
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8
votes
Accepted
Do the terms of an iid sequence whose law has infinite expected value necessarily exceed the...
In 1970, Harry Kesten proposed essentially this question in the Advanced Problems section of The American Math Monthly.
Let $X_1, X_2, \ldots, X_n$ be iid random variables and $S_n = \sum_{i=1}^n …
- 1,136
3
votes
Calculating the probability of an event defined by a condition on a Gaussian random process
I would recommend obtaining a copy of the text
H. Cramer and M. R. Leadbetter (1967), Stationary and Related Stochastic Processes, John Wiley & Sons, Inc.
which is available as a 2004 Dover repr …
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16
votes
Is the function $e^{x^2/2} \Phi(x)$ monotone increasing?
This is just an alternative argument to Davide's nice one.
First, note that $h' = e^{x^2/2}(x \Phi + \Phi')$.
Since $\Phi'' = -x \Phi'$, monotonicity of the integral yields
$$
x \Phi(x) \geq \int_{- …
- 1,136
12
votes
A normal distribution inequality
Yes, the conjectured lower bound is true and can be proved using fairly simple, if somewhat tedious, analysis of derivatives.
First define
$$
b := f - N^2 = x(xN + n) - (xN + n)^2 + N(1-N)\>.
$$
The …
- 1,136