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Results tagged with pr.probability
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user 113397
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
11
votes
Find the area of the region enclosed by $\frac{\sin x}{\sin y}=\frac{\sin x+\sin y}{\sin(x+y...
An approach bypassing polylogarithms is as follows:
\begin{align}\int_1^{1+\sqrt2}\log\frac{t+1}{t(t-1)}\frac{dt}t&\stackrel{ibp}=\int_1^{1+\sqrt2}\frac{t^2+2t-1}{t(t^2-1)}\log t\,dt\\&\stackrel{t=e^u …
- 1,459
0
votes
1
answer
156
views
Analogues of Kac-Bernstein characterisation theorem for non-normal distributions
Let $X,Y$ be two independent random variables.
The Kac-Bernstein theorem states that if $X+Y,X-Y$ are also independent, then $X,Y$ are Normal.
Are there analogues of this theorem for non-normal, conti …
- 1,459
8
votes
3
answers
585
views
Jensen-like inequality for random matrix: $\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$
Let $X\in M_n(\Bbb R)$ be a random matrix with iid elements following a continuous distribution.
What are the necessary and sufficient conditions for $$\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$$ to hold? Is …
- 1,459
10
votes
Accepted
Jensen-like inequality for random matrix: $\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$
From the identities in @OlivierBégassat's answer, the inequality $\Bbb E\det X^2\ge\det\Bbb EX^2$ can be written as $$\small n!\sum_{f=0}^n\binom nf(-1)^{n-f-1}(n-f-1)(\Bbb V[X]+\Bbb E[X]^2)^f\Bbb E[X …
- 1,459
0
votes
A function with unexpectedly simple Legendre transformation
Claim. $J(x)=\tfrac12x^2+\ln|x|+c$ for $|x|>1$ with $c$ constant.
Proof: Here, an explicit form for $I(x)$ is not needed but I can't use it to prove $c=0$.
For $|x|>1$, the substitution $y=2\cos t$ fo …
- 1,459