Questions tagged [expectation]

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Find the Cov$(Z_n^i,Z_n^j)$ where $Z_n^i=\min\{N_i,p\}$, where $N_i$ is binomial

Suppose that $n$ particles (a fixed quantity) are randomly distributed in the interval $[0,1]$ and the particles are independently coming from the uniform distribution in $[0,1]$. Now, consider any ...
33 views

How to calculate the expected distance between two points [closed]

I am trying to calculate the expected distance a mobile user. so if $x_{start}$ and $y_{start}$ are the initial positions and $x_{end}$ and $y_{end}$ are the final positions of the mobile user (which ...
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1 vote
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Expected value of maximal cycle length in fixed-point free bijections

$\newcommand{\n}{\{1,\ldots,n\}}$ $\newcommand{\FF}{\text{FF}}$ $\newcommand{\lc}{\text{lc}}$ Motivation. A group of my son's peers decided to have a few days of Secret Santa before last year's ...
195 views

Statistically stationary properties of expectations conditioned on the value of an Ornstein–Uhlenbeck process

Consider the modified Ornstein–Uhlenbeck process $$\mathop{dx_t}=\theta(y_t-x_t)\, dt+{}\sigma\,dW_t$$ for a standard Brownian motion $W_t$ and $\theta,\sigma\in\mathbb{R}_{>0}$. Let's define the ...
696 views

The expected value of product of random variables which have the same distribution but are not independent

Given a positive integer $k$, is there a positive real number $c(k)$ such that $\mathbb{E}\left(\prod_{i=1}^k X_i\right)\geq c(k)$ for any $k$-random variables $X_1,X_2,\ldots,X_k$ which all have the ...
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A disc contains many random points. Each point is connected to its nearest neighbor. What is the expectation of average cluster size?

A disc contains $n$ independent uniformly distributed points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points. For example, here are $20$ random ...
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Weighted sum of zero-mean random variables

Let us say we have two independent random variables $X$ and $Y$, with both $E[X] = E[Y] = 0$. Is it true that for any random weight variable $0 \le W \le 1$ (e.g., $W$ dependent on $X$ and $Y$) we ...
569 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 ...
• 1,409
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Computing the expectation of a quadratic matrix form involving Bernoulli and Gaussian distributed matrices

I am working with two random matrices, $Z$ and $H$: $Z$ is an $n \times K$ matrix with entries sampled i.i.d. from a Bernoulli distribution: $Z_{ij} \sim \mathrm{Bernoulli}(p)$. $H$ is a $K \times K$ ...
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Tossing a coin around $\mathbb{Z}/n\mathbb{Z}$ [closed]

Motivation. With my younger son I played the following game on a big (dysfunctional) clock which can be modelled as $\mathbb{Z}/12\mathbb{Z}$ : Put the clock hands at number $12 ( = 0)$. Toss a coin, ...
1 vote
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Are there arbitrarily large families of lines in $\Bbb R^3$ with average angle $\ge \pi/3$?

Question: Can I have an arbitrarily large finite family of lines $\ell_1,\dotsc,\ell_n\subset\Bbb R^3$ so that the average angle between two (distinct) lines is $\ge \pi/3$? We can assume that all ...
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116 views

Does $E(XU)\neq 0$ imply $E(f(X) U)\neq 0$ "almost always"?

Consider two non-orthogonal random variables $$(1) \quad E(XU)\neq 0,$$ where $X$ can be a vector. Can we claim that (1) implies that $U$ will be "generically" non-orthogonal to any ...
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1 vote
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Let $(W_t)_{t\ge 0}$ be a standard Brownian motion and $\tau$ be a stopping time lying in $[1,2]$. For $x, y>0$, can we show \mathbb E\big[{\bf 1}_{\{x+\inf_{0\le t\le 2}W_t>0\}}(W_{\tau}-y)^+\... • 1,274 0 votes 1 answer 165 views Union Bound of two events? Inequality 1 \begin{align} \mathbb{P}\left(\frac{1}{n} \sum_{i=1}^{n}\left(f\left(x_{i}\right)-\mathbb{E}[f]\right) z_{i} \geq \frac{\epsilon}{8}\right) \leq 2 \exp \left(-\frac{\epsilon^{2} n d}{9^{4}... 1 vote 1 answer 233 views The maximum trace of a covariance can be achieved by a discrete random vector? Given a random variable X, satifying P(0\leq X \leq 1)=1, and \mathsf{E}[X^2] = \alpha. We know its maximum variance \text{Var}(X) = \alpha(1-\alpha) achived by a binary random variable P(X =... 2 votes 1 answer 131 views Is \lim_{s \rightarrow 1} E(f(X_s)) = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=1}^N f(k)? Let s>1 be a real number. We look at the zeta probability function / Zipf probability function defined as:P(X = n) = \frac{1}{n^s \zeta(s)} Suppose $f: \mathbb{N} \rightarrow \mathbb{R}$ is ...
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Let $h$ be a random variable and $g(h)$ be a real-valued function of $h$. We know that if h is a real-random variable then: $E_h[g(h)] = \int_{-\infty}^{\infty} f(h) g(h) dh$ where f(h) is the PDF of ...