All Questions
Tagged with pr.probability expectation
90 questions
1
vote
1
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54
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Proving bound on expectation of likelihood ratio involving mixtures
Let $p$ be a Lebesgue density function with infinite support (i.e. $p(x)>0 \forall x\in \mathbb{R}$ and $\int p(x) dx = 1$). Moreover, assume that $p$ is even (i.e. $p(x) = p(-x)$) and unimodal: $p(...
- 13
7
votes
0
answers
264
views
$\lim_{n \to \infty} E[V| W +\frac{1}{n}V ]$ where $W$ and $V$ are independent
Let $V$ and $W$ be independent random variables. Assume that $V$ is standard normal.
We are interested in the following limit
\begin{align}
\lim_{n \to \infty} E[V| W +\frac{1}{n}V ]
\end{align}
...
- 671
2
votes
1
answer
154
views
strict inequality for Fatou's lemma
It is not the well-known form of Fatou's lemma. It is shown as below:
let $g\ge 0$ be continuous. If $X_n$ weakly converge to $X$ then
$$\lim\inf_{n\rightarrow \infty} Eg(X_n)\ge Eg(X)$$
I'd like to ...
16
votes
3
answers
782
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Show there is no positive r.v. $U$ such that $\frac{1}{2} = \frac{\mathbb{E}[U^k 1_{U \ge (k+1)/2 }]}{\mathbb{E}[U^k]}, \, \forall k \in \mathbb{N}_0$
Let $U$ be a non-negative random variable such that for all $k \in \mathbb{N}_0$
\begin{align}
\frac{1}{2} = \frac{\mathbb{E}[U^k 1_{U \ge \frac{k+1}{2} }]}{\mathbb{E}[U^k]}.
\end{align}
In ...
- 671
-1
votes
1
answer
80
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Seating assignment inspired question
Motivation. Recently I stayed at a hotel which had the curious custom to ask their $n$ parties (group of guests, most parties a married couple) which of the $n$ tables they wanted to take. Of course ...
- 48.9k
2
votes
0
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71
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Assumptions Wald's second equation?
Let $(X_n)_{n\in \mathbb{N}}$ be an i.i.d. sequence of random variables and $N$ an $\mathbb{N}_0$ valued random variable. Let $X_1 \in \mathcal{L}^2$ and $N \in \mathcal{L}^1$. Let $S_n := \sum_{i=1}^...
- 261
16
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0
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309
views
Randomized Pascal's triangle: What is the average of all the numbers?
This question was posted on MSE. It received some interesting responses, but no definite answer.
Let's build a variation of Pascal's triangle. We write $1$'s going down the sides, as usual. Then for ...
- 3,527
0
votes
0
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134
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Asymptotics of a ratio on the unit sphere
Let $(a_n)_{n \geq 1}$ be a nonnegative, strictly decreasing sequence with $a_n \to 0$ as $n \to \infty$.
Consider the ratio (for $k \geq n$)
$$
R_{n, k} = \mathbb{E}_{u \sim \text{Unif}(\mathbb{S}^{k-...
- 420
1
vote
0
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36
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Uniform distribution as argument for copula likelihood
I am reading a well-known paper about copulas by Chen and Fan (2006). Specifically, Proposition 4.2 (see attached), in which all the arguments are uniform $U_{t-1}, U_t$. However, when the copula is ...
- 33
3
votes
1
answer
271
views
Expectation on a Polish space
I was wondering, if given a Polish space $X$, and given some probability measure $p$ on $X$, can the expectation of an $X$-valued function be taken? In particular, would the integral
$\int_X x dp$ ...
- 291
-1
votes
1
answer
504
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Minimum of exponential distribution
Consider $n$ independent random variables $𝑋_𝑖\sim\exp(𝜆_𝑖)$ for $I=1\ldots,n$. Let $\lambda = \sum_{i=1}^n\lambda_i$. Of course, the minimum of these exponential distributions has distribution:
$...
- 9
3
votes
1
answer
211
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Statistically stationary properties of expectations conditioned on the value of an Ornstein–Uhlenbeck process
Consider the modified Ornstein–Uhlenbeck process
$$\mathop{dx_t}=\theta(y_t-x_t)\, dt+{}\sigma\,dW_t$$
for a standard Brownian motion $W_t$ and $\theta,\sigma\in\mathbb{R}_{>0}$. Let's define the ...
- 41
9
votes
1
answer
788
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The expected value of product of random variables which have the same distribution but are not independent
Given a positive integer $k$, is there a positive real number $c(k)$ such that $\mathbb{E}\left(\prod_{i=1}^k X_i\right)\geq c(k)$ for any $k$-random variables $X_1,X_2,\ldots,X_k$ which all have the ...
- 99
14
votes
1
answer
1k
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A disc contains many random points. Each point is connected to its nearest neighbor. What is the expectation of average cluster size?
A disc contains $n$ independent uniformly distributed points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points.
For example, here are $20$ random ...
- 3,527
-1
votes
1
answer
148
views
Weighted sum of zero-mean random variables
Let us say we have two independent random variables $X$ and $Y$, with both $E[X] = E[Y] = 0$.
Is it true that for any random weight variable $0 \le W \le 1$ (e.g., $W$ dependent on $X$ and $Y$) we ...
- 367
8
votes
3
answers
595
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Jensen-like inequality for random matrix: $\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$
Let $X\in M_n(\Bbb R)$ be a random matrix with iid elements following a continuous distribution.
What are the necessary and sufficient conditions for $$\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$$ to hold? Is ...
- 1,459
0
votes
2
answers
239
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Computing the expectation of a quadratic matrix form involving Bernoulli and Gaussian distributed matrices
I am working with two random matrices, $Z$ and $H$:
$Z$ is an $n \times K$ matrix with entries sampled i.i.d. from a Bernoulli distribution: $Z_{ij} \sim \mathrm{Bernoulli}(p)$.
$H$ is a $K \times K$ ...
- 37
0
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0
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46
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Prove lower collinearity on the tails of Gaussian blob
Let us consider a $n$-dimensional Gaussian blob, i.e. a set of $N$ random vectors $\{\boldsymbol{X}^{(j)}\}_{j=1}^N$, with $n$ independent components, $X_i^{(j)}$, and such that $X_i^{(j)} \sim \...
- 305
5
votes
1
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466
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Bounding expected maximum via adjacent differences
For $X_i$, $i\in[n]$
be a sequence of integrable random variables.
Is there a universal constant $c>0$ such that
$$\mathbb{E}\max_{i\in[n]}X_i
\le
c\left(
\max_{i\in[n]}\mathbb{E}|X_i|
+
\mathbb{E}\...
- 6,541
5
votes
2
answers
166
views
Expectation of a function of two entries of an isotropic unit vector $\mathbb{E}_{\mathbf{w}\sim\mathcal{S}^{p-1}}\![{w_{1}}^{\!4}\,{w_{2}}^{\!4}]$
Problem:
Given a random isotropic unit vector in $\mathbb{R}^p$ for $p\ge2$, we are trying to compute (preferably exactly, otherwise to upper bound):
$$\mathbb{E}_{\mathbf{w}\sim\mathcal{S}^{p-1}}\!\...
- 673
0
votes
1
answer
113
views
Inequality on conditional variance of a vector
I have a random vector $X$ and an event $\mathcal{E}$ such that $\mathbb{P}(\mathcal{E}) = p$. I am trying to show the following inequality :
\begin{equation}
p\mathbb{E}[\|X - \mathbb E [X \vert \...
- 11
6
votes
2
answers
293
views
Expectation of the inner product of a subset of two random orthonormal vectors
Setting: Consider sampling two orthonormal vectors $\mathbf{u},\mathbf{v}$ in $\mathbb{R}^p$ (where $p\ge2$) from a "uniform" distribution over the $p$-dimensional sphere (alternatively, ...
- 673
13
votes
2
answers
398
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Expected sorting time of random permutation using random comparators
In sorting networks, a comparator of positions $i < j$ is an operator which takes a permutation, checks if $p_i > p_j$, and if it is the case, swaps $p_i$ and $p_j$.
Using this, we can define ...
- 3,319
2
votes
1
answer
194
views
Bound the probability that a point belongs to a set
Let $(a_k)_{k \geq 1}$ be random variables taking values on a finite subset $B$. Assume that
$$
(1) \quad \Pr\Big (\lim_{n\rightarrow +\infty}d(\frac{1}{n}\sum_{k=1}^n 1_{[a_k = b]}, [v_\ell(b,\...
- 108
0
votes
1
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197
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Bound the expectation of an average
Let $(a_n)_{n \geq 1}$ be random variables taking values on a finite subset $B$. Assume that $\nu_l(b) \le P[a_n = b\mid a_1,\ldots,a_{n-1}] \le \nu_u(b)$ almost surely for every $n \ge 1$ and $b \in ...
- 108
2
votes
1
answer
196
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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, ...
- 48.9k
1
vote
1
answer
204
views
Expected (maximum minus minimum) of Laplacian random variables
Suppose there are $n$ IID random variables denoted as $X=(X_1,\dots, X_n)$, they follow Laplace distribution with parameter $\lambda$, denoted as $Lap(\lambda)$. That is,
$$f(x)=\frac{1}{2\lambda}\exp ...
- 23
2
votes
1
answer
324
views
On the mean value taken by Bernoulli random variables with joint distribution constraints
We are given a vector $n$-dimensional random vector $\mathbf{X}$ whose components are the Bernoulli random variables $X_1, X_2, \ldots X_n$, such that the probability $\mathbb{P}(X_1=X_2=\ldots=X_n=0)$...
- 1,677
0
votes
1
answer
115
views
Approximation for an expectation expression
Let $\mathbf{x} \in \mathbb{C}^M$ is an unknown distributed random vector (certainly not gaussian), and matrix $\mathbf{A}\in \mathbb{C}^{M \times M}$ which is fix (known). Also, assume we know the ...
- 25
31
votes
5
answers
2k
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On average, how many uniformly random real numbers $u$ are needed for their sum to exceed $1$, if $u_1$ is in $(0,1)$ and $u_k$ is in $(0,eu_{k-1})$?
A well-known question is: on average, how many uniformly random real numbers in $(0,1)$ are needed for their sum to exceed $1$? The answer is $e$.
Let's tweak this question by making each random ...
- 3,527
1
vote
0
answers
168
views
Expectation of inverse of complex Gaussian variables
If we consider a complex Gaussian random variable as $h\sim\mathcal{CN}(0,\gamma)$, where $\gamma$ is the variance. Is there any closed-form solution with $\gamma$ for $\mathbf{E}\left[\frac{1}{\lVert ...
- 11
1
vote
1
answer
198
views
Probability of multivariant gaussian random variables in different areas
$\newcommand{\sgn}{\operatorname{sgn}}$Let $X_i$ is a gaussian random variable correlated with others. we want to find the probability of each possible case to find the expectation of following ...
- 25
0
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1
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184
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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}...
user158501
2
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1
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138
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)}$$
Suppose $f: \mathbb{N} \rightarrow \mathbb{R}$ is ...
- 3,064
1
vote
0
answers
19
views
Сonditional characteristics with respect to a discrete random variable [closed]
160 asymmetrical coins participate in the first roll. In the second roll, only those coins on which the "eagle" fell out in the first roll participate. It is known that the probability of an ...
- 11
0
votes
1
answer
243
views
Integral form of expectation with respect to complex random variables [closed]
Let $h$ be a random variable and $g(h)$ be a real-valued function of $h$.
We know that if h is a real-random variable then:
$E_h[g(h)] = \int_{-\infty}^{\infty} f(h) g(h) dh$ where f(h) is the PDF of ...
- 13
3
votes
2
answers
593
views
A lower bound for the expectation of $\min\{X,n-X\}$ when $X$ follows a $\mathrm{Binomial}(n,p)$ distribution
Let $X$ be a random variable following a $\mathrm{Binomial}(n,p)$ distribution, and let $$Y=\min\{X,n-X\}.$$ Ispired by the problem posed by C. Clement on https://math.stackexchange.com/questions/...
- 377
7
votes
1
answer
347
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Expectation for game choosing uniformly number in $[0,1]$ until it decreases
We are playing a game where we keep on choosing a number from the uniform distribution U(0,1). The game goes on until we have the current number less than the previously picked number, i.e. the game ...
2
votes
1
answer
274
views
Inequality for increments of $r$th absolute moments of martingales, $1<r<2$
If $Y_n=\sum_{i=1}^n X_i$ is a martingale, where $X_i$ is a martingale difference sequence, $\mathbb{E}[X_n\mid \mathcal{F}_{n-1}]=0$ for all $n$, we know that
$$ \mathbb{E}\big[Y_n^2-Y_{n-1}^2\big]=\...
- 23
2
votes
1
answer
387
views
Expected value of attempts needed to find a "pair" of cards
We are given an integer $n \geq 1$ and $2n$ cards, labelled $0$ to $2n-1$. We pick a card with uniform probability, put it back, and continue, until for some $k\in \{0,n-1\}$ the cards
$2k$ and $2k+1$ ...
- 48.9k
1
vote
0
answers
81
views
Calculating the mean squared error for an estimate of a large sum
Consider the set of all Boolean function $f: \{0, 1\}^{n} \rightarrow \{-1, 1\}$. Now, let's pick a function uniformly at random from this set. Let $F$ be the random variable corresponding to the ...
0
votes
2
answers
963
views
Product of three or more independent sub-Gaussian varibles
A random variable $X$ is called subgaussian of order $\sigma^2$ if $\log E[exp\{\theta X\}]\leq \frac{1}{2}\theta^2\sigma^2$ for every $\theta\in\mathbb R$.
Given a sequence of independent subgaussian ...
- 59
2
votes
0
answers
158
views
Size of an “average” ϵ-net on the unit sphere
This is a question I originally asked on math.stackexchange, but didn't receive a satisfying answer.
Let $\epsilon>0$ and consider constructing a set $S_\epsilon\subseteq S^{d-1}$ of points on the ...
- 618
1
vote
0
answers
157
views
Pulling random times out of conditional expectation ("Substitution rule")
Problem
Let $G$ be a positive random variable (a random time) that is a.s. finite, $(X)_{t \geq 0}$ be a càdlàg process taking values in $\mathbb{R}^d$ and $g$ is some sufficiently nice real-valued ...
1
vote
0
answers
121
views
Relation satisfied by a Gaussian random variable
I want to prove the following relation for $X\sim \mathcal{N}(0,1)$, $x\in \mathbb{R}$ and $f(x)=\mathbb{E}[\max(X,x)]$:
$$f(\frac{f(x+1)+f(x-1)}{2})\leq \frac{f(f(x)-1)+f(f(x)+1)}{2}$$
It seems that ...
- 171
1
vote
1
answer
141
views
Expectation of maximum of all period lengths of functions $f:\{1,\ldots,n\}\to\{1,\ldots,n\}$
This is based on an older question.
For $n\in\mathbb{N}$, let $[n]:= \{1,\ldots,n\}$. Let $\text{Fun}(n)$ denote the set of all functions $f:[n]\to[n]$. To $f\in\text{Fun}(n)$ and ''starting value'' $...
- 48.9k
6
votes
2
answers
274
views
Expectation of period length of functions $f:\{1,\ldots,n\}\to \{1,\ldots,n\}$
For $n\in\mathbb{N}$, let $[n]:= \{1,\ldots,n\}$. Let $\text{Fun}(n)$ denote the set of all functions $f:[n]\to[n]$. To $f\in\text{Fun}(n)$ associate a sequence $\text{seq}(f))$ defined recursively by
...
- 48.9k
2
votes
1
answer
85
views
Judge a special positive definite matrix in probability
Assume $\mathbf{x}$ is a random vector. The question is to judge whether
$$E \{ (\mathbf{xx'})^{-1} \}- E\{(\mathbf{xx'})\}^{-1}$$
is positive definite or not.
I have no idea how to do it. Could ...
- 51
8
votes
3
answers
628
views
Expected distance between two uniform points in distinct rectangles
Are there any good approximations (especially upper bounds) for the quantity $E(\|X_1-X_2\|$), where each $X_i$ is uniformly distributed in a rectangle $[a_i,b_i]\times[c_i,d_i]$? It does not appear ...
- 4,049
4
votes
1
answer
222
views
Memory game inspired problem
Motivation. As I was playing the pairs-matching game "Memory" (known as "Concentration" in some parts of the world) with my children, I was surprised that even thorough shuffling ...
- 48.9k