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2
votes
0answers
43 views

The expected size of a subtree of any labelled rooted tree

Consider the set of labelled rooted trees of size $n$, $\mathcal{T}_n$. Let $r$ be the root of each tree $T=(V,E)\in\mathcal{T}$, $r\in V(T)$, and let $n(u)$ be the number of vertices of the subtree $...
1
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2answers
104 views

Disintegration, conditional probabilities, and conditional expectation

On the Wikipedia page there is a note that conditional probability measures can be described by disintegration. However, I can seem to find a clear exposée of how this construction is related to ...
0
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0answers
85 views

A minimax optimization with expectation operator

Let $\mathbf{X}$ be an $m\times n$ random matrix with Gaussian i.i.d entries with zero mean and unit variance. How we can think about the following optimization \begin{align} \max_m\mathbb{E}_\mathbf{...
1
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1answer
26 views

An attempt to find expected value of clique number of special random graph

Let $G(n)=(V,\mathcal{E})$ be a random graph definded as follows: $V=[n]=\{1,2, ... ,n\}$ and for all $i,j\in V$ so that $i\ne j$ we have $\{i,j\}\in\mathcal{E}$ with probability $p$. Where $p\in[0,1]...
4
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1answer
177 views

Expectation of multiplied random variables given their individual expectations

Suppose that I have two non-negative real valued random variables $x, y \in Z_+$ that always satisfy $$x+y \leq 1.$$ Also suppose that $E[x] = 1/2$ and $E[y] = 1/4$. What is the maximum possible value ...
-2
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1answer
98 views

In which cases $E(e^{t S_n S_m})$ converges to $E(e^{t X Y})$

Suppose that $S_n$ and $S_m$ are two random binomial variables, which are independent and with the same distribution parameter $p$. I am wondering, in which cases $E(e^{t S_n S_m})$ converges to $E(e^...
-1
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1answer
62 views

If a sequence $X_n$ of RVs converges in probability to $X$, does the sequence $\mathbb{E}(X_n)$ also converge to $\mathbb{E}(X)$? [closed]

I couldn't find the answer in literature so any idea would be helpful.
0
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0answers
41 views

Expected value of minimal shift in permutations

The following occurred to me when playing a game involving shuffling a deck of cards with my children. Let $S_n$ denote the set of bijective maps (permutations) $\pi:\{1,\ldots, n\}\to\{1, \ldots, n\}...
0
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0answers
21 views

Approaches and approximations for specific non-convex low-rank optimization problem

I have recently begun looking at an optimization problem in the context of Wiener filtering and lower-dimensional representations. Assume $ X $ is a random vector of length $ n $ and $ Y $ is a random ...
2
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0answers
42 views

An upper bound on $\mathbb{E}\bigg[\bigg(\sum_{i=1}^{k}(X^{\top}A_{i}X)^{2}\bigg)^{q}\bigg]$

Let $X\in\mathbb{R}^{d}$ have independent, mean zero subgaussian entries, and $A_{1},\ldots,A_{k}$ be fixed $d\times d$ matrices that have zeros on the diagonal. I would like to upper bound the ...
1
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1answer
53 views

Conditional density for random effects prediction in GLMM

I am currently working on generalized linear mixed models (GLMM) and need some help concerning the prediction of the random effects. More specifically, I don't understand the given representation of ...
4
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2answers
455 views

Bounding an expectation involving i.i.d. standard Gaussians and Rademacher

I have tried to bound the following quantity, but cannot get the "right" (conjectured) bound: $$ \phi(\gamma,d,n) = -1+e^{\frac{1}{2}n\gamma^2 d} \mathbb{E}_{X}\left[\frac{\mathbb{E}_Z[\prod_{j=1}^n(...
0
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0answers
37 views

moment generation function for matrix Gaussian distribution

What is the moment generation function for the following function $$ E(e^{\mathbf{X}^T\mathbf{y}\mathbf{W}})=\int e^{\mathbf{X}^T\mathbf{y}\mathbf{W}}\frac{\exp\big(-\frac{1}{2}\mathrm{tr}\big[\mathbf{...
0
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1answer
103 views

Bounds on expectation of X/(X^2 + c) with X ~ Gaussian and c > 0

I'm trying to compute expectation of $X / (X^2 + c)$ when $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$, and $c$ is some positive constant. I think this cannot be solved ...
0
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0answers
75 views

Moment of a function of Gaussian random variables: $\mathbb{E}[(a_{i}^{\top}AA^{\top}a_{j})^{q}]$

Let $A$ be an $m\times k$ matrix with iid $\mathcal{N}(0,1)$ entries and $a_{i}$ and $a_{j}$ be its $i$th and $j$th columns, respectively. I would like to compute the following quantity: \begin{...
0
votes
1answer
34 views

expectation of a quadratic function of a matrix variate normal distribution

I want to compute the following expectation term: $E[{\bf{XA}}{{\bf{X}}^T}]$ where ${\bf X} \in R^{M \times M}$ and its elements are normal random variables such that $vec\left( {\bf{X}} \right)\...
0
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1answer
82 views

Are there known bounds on the expectation of the truncated Beta distribution?

Let $X\sim beta(\alpha,\beta)$ be a random variable and let $\tau\in(0,1)$. Are there any known closed-form bounds (I'm specifically interested in lower bounds) on $$ \mathbb E[X\ | X\le \tau]? $$
1
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1answer
97 views

upper bound of the expectation of upper quantile ratio

We have a collection $\boldsymbol{S}$ of $n$ discrete random variables $X_1$, $X_2$, $\dots$, $X_n$ $\overset{\small \text{i.i.d.}}{\small \sim}$ $\mathcal{D}$, where $\mathcal{D}$ is a distribution ...
-1
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1answer
120 views

expectation of upper quantile proportion

(edited considerably following comments) We have a collection $\boldsymbol{S}$ of $n$ discrete random variables $X_1$, $X_2$, $\dots$, $X_n$ $\overset{\small \text{i.i.d.}}{\small \sim}$ $\mathcal{D}$...
0
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0answers
70 views

The conditional expectation of X_1^2 under summation X_1 + … + X_n

Given $x_1, x_2, \ldots, x_n$ that are i.i.d. with mean 0 and variance 1. What is the conditional expectation of $$ \mathbb{E}[x_1^2|x_1+\cdots+x_n] $$ I was trying to think that $\mathbb{E}[x_1^2|...
3
votes
1answer
132 views

Expected value of “longest bit / shortest bit” in $n$ uniformly distributed points on $[0,1]$

Let $n\geq 2$ be an integer. We pick $n$ points in $[0,1]$ with uniform distribution. Let $A$ be the minimum distance that two adjacent points have, and let $B$ be the maximum distance that two ...
1
vote
2answers
62 views

Expectation of $\left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textbf{y} \|^2} \right|^2$, with complex Gaussians?

Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ...
7
votes
1answer
323 views

Expected value of biggest distance of adjacent points uniformly picked in $[0,1]$

We pick $n\ge 2$ points in $[0,1]$ with uniform distribution. What is the expected value of the largest distance of $2$ adjacent points?
0
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0answers
69 views

A closed form of mean-field equations

Assume that a system at time t, for example number of costumers in a line at time $t$ which is denoted by $q(t)$, follows a Markov chain with these dynamics (probabilities) $$P(q(t+\Delta t)-q(t)=1)=\...
0
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0answers
51 views

Root of the expectation of a random rational function

I am trying to figure out a formula for the unique $\lambda>1$ such that $$ \mathbb{E}\bigg[\frac{X}{\lambda -X}\bigg]=1 $$ where $X$ is a discrete random variable taking values in $\{\frac{1}{n},....
1
vote
1answer
87 views

The expectation of binary logistics regression with respect to Gaussian distribution

I am trying to compute the expectation of $g(s,x)=s \ln \sigma(x)+(1-s)\ln(1-\sigma(x))$ with respect to the normal distribution $\mathcal{N}(x;m,v)$, where we have $\sigma(x)=\frac{1}{1+e^{-x}}$. If ...
1
vote
1answer
226 views

Expectation of a linear operator

We define $T: C[0,1]\to C[0,1]\ni T(f(x))= \sum\limits_{k=1}^{m} p_k (f\circ f_k)(x):=\mathbb E( f(X_{n+1}|X_n=x)$ for a system $X_{n+1}=f_{\omega_n}(X_n), n=0,1,2\dots.$ and $\omega_n$ are i.i.d ...
2
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2answers
155 views

Substitute Concrete Value in Conditional Expectation

Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space. Let $$ X, Y : \Omega \rightarrow \mathbb{R} $$ be random variables. Furthermore, let $$ f: \mathbb{R}^2 \rightarrow \mathbb{R} $$ be a $...
1
vote
1answer
265 views

Is there a tight lower bound for the expectation of the product of two positive valued random variables?

Let $X,Y$ be two (dependent) random variables with $\mathbb{P}(X\ge 0)=\mathbb{P}(Y\ge 0)=1$. I want to find a tight lower bound of $\mathbb{E}(XY)$ when $X,Y$ are non-negative, almost surely. ...
4
votes
2answers
169 views

Expected value of length of interval game

I have a die that produces uniformly distributed values in $\{1,\ldots, k\}$ for some integer $k\geq 2$. Now I play the following game. I start rolling the die and produce one integer in $\{1,\ldots,...
0
votes
1answer
63 views

CDF of a RV that is the ratio between a complex Gaussian and a Chi-squared RVs

Given the following p.d.f., which is the p.d.f. of the real and imaginary parts of a random variable that is the ratio between a complex Gaussian and a Chi-squared RVs: \begin{equation*} f_U(u)=\exp\...
1
vote
1answer
79 views

Minimization Proof of Conditioning on Gaussian is Gaussian

It is well known that $E[X|X+Y]$ is Gaussian if both $X$ and $Y$ are, and the result can be derived using standard density arguments. However, how can one prove it by only resulting to optimization ...
2
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2answers
130 views

Expectation of $\left| \frac{\textbf{x}^{H} \textbf{y} }{\| \textbf{x} \|^2} \right|^2$, where $\textbf{x}$ and $\textbf{y}$ are complex Gaussians?

Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ...
2
votes
1answer
82 views

p.d.f. of $\left| \frac{\textbf{x}^{H} \textbf{y} }{\| \textbf{x} \|^2} \right|^2$, where $\textbf{x}$ and $\textbf{y}$ are complex Gaussians?

Given that the random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ are ...
22
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5answers
1k views

Maximizing the expectation of a polynomial function of iid random variables

Let $f \colon \mathbb R^N \to \mathbb R$ be a smooth function. Let $\mu$ be a probability measure on $[0,1]$ and $X_1, \ldots , X_N$ be i.i.d. random variables on $\mathbb R$. Question 1. What is ...
2
votes
2answers
223 views

Closed expression for $\mathbb{E} \left\lbrace \Re \frac{(\textbf{x} + \textbf{y})^{H}\textbf{x}}{\| \textbf{x} + \textbf{y}\|^{2}} \right\rbrace$?

Given the random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ are independent, ...
4
votes
1answer
265 views

How sensitive are probability distributions to noise?

I'm trying to prove a result but I'm stuck at the very end of it: I'm having troubles understanding how noise propagates when considering a probability distribution. In other words, if I inject some ...
0
votes
1answer
193 views

Conditional Expectation given integral of a Brownian motion

I'm trying to find $$E \left[\int_3^4 B_t \, dt \mid \int_1^2 B_t \, dt= c\right]$$ where $B_t$ is standard Brownian motion and integral is just a Riemann integral and constant $c$ is a known real ...
1
vote
0answers
79 views

If $\text P\left[X_2\in B_2\mid X_1\right]=\kappa(X_1,B_2)$ a.s. for all $B_2$, can we select a common null-set over all $B_2$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(E_i,\mathcal E_i)$ be a measurable space $X_1:\Omega\to E_1$ $X_2:\Omega\to E_2$ be $(\mathcal A,\mathcal E_2)$-measurable $\kappa$ ...
6
votes
1answer
291 views

Approximating $\mathbb{E}[1/X]$

I am well aware (as for instance discussed here https://math.stackexchange.com/questions/910846/is-it-true-in-general-that-e1-x-1-ex) that for an arbitrary random variable $X$ it does not hold that $\...
6
votes
2answers
166 views

Neighboring number of a permutation

For any positive integer $n\in\mathbb{N}$ let $S_n$ denote the set of all bijective maps $\pi:\{1,\ldots,n\}\to\{1,\ldots,n\}$. For $n>1$ and $\pi\in S_n$ define the neighboring number $N_n(\pi)$ ...
3
votes
2answers
115 views

What is $\sum_{k=0}^{+\infty}{k⋅p(k;\mu_1,\mu_2)}$, where $p$ is the pmf of Skellam distribution?

The Skellam distribution is the discrete probability distribution of the difference $N_{1}-N_{2}$ of two statistically independent random variables $N_{1}$ and $N_{2}$, each Poisson-distributed with ...
2
votes
0answers
58 views

Second moment of ranks

Suppose vector $R$ is a random permutation of the integers 1 through $n$ such that $$ \mathcal{P}\left(R_i = 1\right) = \pi_i, $$ for given vector of probabilities $\pi$. Moreover, assume a '...
5
votes
1answer
157 views

Сoincidence of discrete random variables

Let $\xi, \eta$ be a discrete random values and $\mathbb E| ξ |$, $\mathbb E | η | < +\infty$, and any value of these values ​​are accepted with a non-zero probability. How to prove that from $\...
2
votes
0answers
100 views

Counterexample in Kolmogorov theorem about existence of almost surely continuous modification

I want to understand this Kolmogorov theorem about existence of almost surely continuous modification: A process $\{\xi_t, \in[0,T]\}$ admits an almost surely continuous modification if there exist ...
4
votes
3answers
131 views

Expected distance of nearest matching pair in the game of pairs

Recently I was playing several rounds of the game of pairs with my children. I was surprised that almost every time, one matching pair was adjacent (either next to each other in a row, or vertically). ...
5
votes
1answer
307 views

Expectation of exponential of a function of independent Rademacher r.v.'s involving the error function

Let $Z,Z'\in\{-1,1\}^n$ be two independent vectors of i.i.d. Rademacher r.v.'s, where $1\leq n \leq d$ are two integers ($d\gg 1$). I am trying to get an upper bound on $$ \mathbb{E}_{ZZ'}\left[ \exp\...
1
vote
0answers
148 views

Expected value of inverse of complex non-central Wishart matrix

I have a matrix $W$ that abides a complex non-central Wishart distribution. My question is what the expectation of the inverse is, i.e., how to compute $$\mathbb{E}(W^{-1}).$$ I have tried to read up ...
1
vote
2answers
227 views

lower bound the probability of at least L collisions

Lets say we get a list $M$ containing $|M|=\sqrt{L\cdot N}$ randomly and independtly drawn elements from a set of size $N$. And lets denote the $i$-th element of the list $M$ by $M[i]$. If we now ask ...
1
vote
1answer
113 views

On the eigenvalue of the expectation value of a random matrix in quadratic form

When we handle with some dynamic input-output mappings, there occurs a question as follows: Let $M$ be a random matrix, of which each element contains random terms. Consider the two expectation ...

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