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The proof of $$ \mathbb E \Big| \sum_{i=1}^{2n} x_i^2 - \sum_{j=1}^{2n} y_j^2 \Big| = 4^{1-n}n \binom{2n}{n}. $$ Denote $z_i=(x_i-y_i)/\sqrt{2}$, $w_i=(x_i+y_i)/\sqrt{2}$. Then the vectors $z$ and …

This is possible for all $n$ and $p$. I start with a direct construction. Obviously, if $X$ is even, then we should have $X'=X$. So we should construct the corresponding coupling between $Y$ and $X'$, …

$\sqrt{\varepsilon}$ works. Assume that for some set $A$ we have $\mu(A)=a$ and $\nu(A^{\sqrt{\varepsilon}})<a-\sqrt{\varepsilon}$. Then with probability more then $\sqrt{\varepsilon}$ we have $X\in A …

(Something to start with.) Denote the permanent by $P$. We have ${\mathbb E}(P)=(1-p)^nn!$. Now look at ${\mathbb E}(P^2)$. This is a sum over all pairs of permutations $\pi,\sigma$ of $(1-p)^{2n-fix …

Yes, this is called 'majorization' or 'second order stochastic dominance' of measures (first term is used in analysis, second in probability). The idea is very simple: we partition the measure $\mu$ o …

We may change the variables as $xa^{1/s}=t$, this proves that $a^{1/s}\cdot \int$ is a function of $ba^{-p/s}$, not of $ba^{1/s}$.

The probability that all samples are less than $N^{19/10}$ is $(1-N^{-19/20})^{N}$ that tends to 0. The expected number of samples greater than $N^{1/2}$ is $N^{3/4}$, thus, the probability that we ha …

The distribution function equals $$ p(x)=\frac{|x-a|}{(a-b)(a-c)}+\frac{|x-b|}{(b-a)(b-c)}+\frac{|x-c|}{(c-b)(c-a)}. $$ This is pretty symmetric. If you need a $k$-th moment, it equals $\frac2{(k+1)( …

As Paata suggests, we write $$ \mathbb{E} e^{tZ^{2}} = 2t \int_{0}^{\infty}\lambda e^{t\lambda^{2}}P(Z>\lambda)d\lambda. \quad\quad\quad (\heartsuit) $$ Next, for any vector $(\delta_1,\ldots,\delta_n …

Yes, it is a part of Fubini theorem for the characteristic function of $Y$.

Denote $I_0(n)=\int_0^r x^{nr-1}(1-x)^{1+(1-r)n}dx$, $I_1(n)=\int_r^1 x^{nr-1}(1-x)^{1+(1-r)n}dx$. You want to prove that $I_0/(I_0+I_1)$ decreases, equivalently, $(I_0+I_1)/I_0$ increases, equivalent …

If we denote the density $d\mu/dx=p(x)$ (considered as a function on the whole line), for the function $f(L)=\hbar_{\nu,L}$ (also considered on the whole line) we get $f(L)=-L$ whenever $L<0$ and $$f …

Denote $K_0=[-a,a]\times \mathbb{R}^{k-1}$, $K_{-}=[-\infty,-a]\times \mathbb{R}^{k-1}$, $K_{+}=[a,\infty]\times \mathbb{R}^{k-1}=-K_-$. We have $$\mu(L)(\mu(K_0)+2\mu(K_+))=\mu(L)=\mu(L\cap K_0)+\mu( …