Questions tagged [expectation]
The expectation tag has no usage guidance.
44 questions with no upvoted or accepted answers
16
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309
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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 ...
- 3,507
7
votes
0
answers
263
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
6
votes
0
answers
164
views
Stopping time on the sum of a moving window
(Cross post from MSE)
Let $\xi_i$ be iid random variables, and define:
$$S_{(k)} = \sum_{i=1}^N \xi_{i+k}$$
Now, define:
$$\tau = \min \left\{ k : S_{(k)} \notin (a,b) \right\}$$
How can I find $\...
- 193
4
votes
0
answers
249
views
A tricky integral equation
In the context of reconstruction of climate data from ice cores (see related question at MSE) I came about the following problem. (More background and motivation can be given on demand.)
Let $f:\...
- 9,720
3
votes
0
answers
63
views
How to prove emprical risk converges to expectation risk as $n\to \infty$?
For example, for a classical binary classification:
$x \in \mathbb{R}^d$ and $y \in\{0,1\}$
let empirical risk be
$R_{\ell}^n(f):=\frac{1}{n} \sum_{i=1}^n \ell\left(f\left(X_i\right), Y_i\right)$
and ...
- 51
2
votes
0
answers
71
views
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
2
votes
0
answers
74
views
Expected number of steps until a queue of $n$ people has passed all $n$ ordered tests consecutively
We are given a queue of $n$ people $\{p_1, \ldots, p_n\}$. They each have to pass $n$ exams $\{t_1, \ldots, t_n\}$. For simplicity we can "draw" the setting in the following way:
$$[t_n,t_{n-...
- 21
2
votes
0
answers
122
views
Are the extensions of the expected value, below, finite for all functions in only a shy subset of all measurable functions?
This is a follow up to this post, where I wish to verify whether one of the statements (in the post) is true but first let's recap the definitions:
Let $(X,d)$ be a metric space. If set $A\subseteq X$,...
- 63
2
votes
0
answers
42
views
can we get a family of classifiers $\left\{f_n\right\}_{n \in N}$such that $\lim_{n->∞} (E_{(X_1, Y_1), ...,(X_n, Y_n) \sim \rho}[R(f_n)]-R(f_B))=0 $
For a given classifier $f: \mathbb{R}^d \mapsto\{0,1,2\}$, let
$$
R(f):=\mathbb{E}_{(X, Y) \sim \rho}\left[\mathbb{1}_{f(X) \neq Y}\right]
$$
$f_B$ the Bayes classifier.
can we get a family of ...
- 51
2
votes
0
answers
730
views
Expected value of length of longest cycle in permutation
Let $n$ be a positive integer and let $S_n$ be the collection of permutations $\pi:\{1,\ldots,n\}\to \{1,\ldots,n\}$. For $\pi\in S_n$ let $\text{maxcyc}(\pi)$ denote the length of the longest cycle ...
- 48.8k
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
2
votes
0
answers
336
views
For the following class of matrices, are the determinants invariant under permutations?
I want to ask a question regarding the invariance of determinants under permutation. The following matrix is the one I want to discuss here. (It's just a symmetric block tridiagonal matrix with non-...
- 21
2
votes
0
answers
55
views
Expectation of Hadwiger number of a random graph
For any integer $n$, let ${\cal G}_n$ denote the set of simple, undirected graphs $G = (V, E)$ where $V = \{1,\ldots,n\}$. The Hadwiger number $\eta(G)$ of a finite graph $G$ is the maximum integer $m$...
- 48.8k
2
votes
0
answers
112
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 $...
2
votes
0
answers
58
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 ...
- 129
2
votes
0
answers
61
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,
$$
for given vector of probabilities $\pi$.
Moreover, assume a '...
- 620
2
votes
0
answers
244
views
Counterexample in Kolmogorov theorem about existence of almost surely continuous modification
I want to understand this Kolmogorov theorem about existence of almost surely continuous modification:
A process $\{\xi_t, \in[0,T]\}$ admits an almost surely continuous modification if there exist ...
- 33
1
vote
0
answers
47
views
Equality regarding score function
$\newcommand{\Var}{\operatorname{Var}}$I'm reading Bogetoft and Otto (2011) page 249 and this equality regarding the score function stumped me:
$$\Var\left[ -\Big(\frac{\partial^2 l}{\partial \beta^2}\...
- 11
1
vote
0
answers
36
views
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
1
vote
0
answers
107
views
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 ...
- 48.8k
1
vote
0
answers
739
views
Finding a unique and finite expected value for almost all measurable functions?
Let $(X,d)$ be a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,+\infty)$ and $\text{dim}_{\text{H}}(A)$ is the ...
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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
0
answers
297
views
Expected value of ceiling of a random variable
I have a continuous non-negative random variable $X \ge 0$ defined by a black-box cumulative distribution function $F(x) = \Pr [ X \le x ]$. In other words, I have an algorithm to calculate $F(x)$ for ...
- 367
1
vote
0
answers
2k
views
What is known about the maximum of several independent standard normal random variables?
Bailey et al., in the May 2014 issue of Notices of the AMS, show that, if $X_k\,$ ($k=1,...,n$) are independent standard normal random variables, then the expectation of their maximum is given ...
- 2,437
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 ...
1
vote
0
answers
156
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
0
answers
82
views
Negative moments of Steinhaus random variables
Let $f_i, i=1, \ldots, n$ be independent Steinhaus random variables, i.e. variables which are uniformly distributed on the complex unit circle. Let $a \in R^n$.
1) Find $E\left(\sum_{i=1}^nf_i a_i\...
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1
vote
0
answers
81
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
$\kappa$ ...
- 167
1
vote
0
answers
269
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 ...
- 11
1
vote
0
answers
132
views
Mean and correlation of product of two random processes
I have two random process:
$$A(at)$$
$$\cos(2\pi f_0t+\Phi)$$ with these hypothesis:
$a$ and $f_0$ are constant
$\Phi$ is uniformly distributed in $[0,\pi)$
$A(at)$ is WSS
I must calculate the ...
- 111
1
vote
0
answers
73
views
If $f$ is a measurable random field, then $(ω,x)↦E[f(x)\mid F](ω)$ has a measurable version $g$ and $E[f(X)\mid F]=g(X)$ for all $F$-measurable $X$
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$\mathcal F\subseteq\mathcal A$ be a $\sigma$-algebra on $(\Omega,\mathcal A)$
$(E,\mathcal E)$ be a measurable space
$f:\Omega\...
- 167
0
votes
0
answers
133
views
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 $k \geq n$)
$$
R_{n, k} = \mathbb{E}_{u \sim \text{Unif}(\mathbb{S}^{k-...
- 460
0
votes
0
answers
37
views
Expected (log)volume of projection of a cube onto a random subspace
Suppose $A$ is a full-rank $d\times d$ dimensional matrix. Let $U \in R^{d\times k}$ be a projection to projection matrix onto a uniformly chosen sub-space of dimension $k$ (for example, they can be ...
- 187
0
votes
0
answers
46
views
Prove lower collinearity on the tails of Gaussian blob
Let us consider a $n$-dimensional Gaussian blob, i.e. a set of $N$ random vectors $\{\boldsymbol{X}^{(j)}\}_{j=1}^N$, with $n$ independent components, $X_i^{(j)}$, and such that $X_i^{(j)} \sim \...
- 305
0
votes
1
answer
113
views
Inequality on conditional variance of a vector
I have a random vector $X$ and an event $\mathcal{E}$ such that $\mathbb{P}(\mathcal{E}) = p$. I am trying to show the following inequality :
\begin{equation}
p\mathbb{E}[\|X - \mathbb E [X \vert \...
- 11
0
votes
0
answers
84
views
Expectation of the trace of random matrix with an inverse insided
Consider a $N$-dimensional random complex vector $\mathbf{x} \in \mathbb C^{N \times 1}$ following the complex Gaussian distribution, i.e., $\mathbf{x} \sim {CN}(0,\sigma^2 \mathbf{I})$, where $\...
0
votes
0
answers
171
views
How many elements have a "small" order in a finite field?
I'm hoping that this is an easy question for someone.
How many elements can we expect to have multiplicative order at most $n^{1/c}$ in one of the finite fields $\mathbb{F}_p$ with $p$ prime with $n \...
- 221
0
votes
0
answers
127
views
Does $E(XU)\neq 0$ imply $E(f(X) U)\neq 0$ "almost always"?
Consider two non-orthogonal random variables
$$
(1) \quad E(XU)\neq 0,
$$
where $X$ can be a vector.
Can we claim that (1) implies that $U$ will be "generically" non-orthogonal to any ...
- 108
0
votes
0
answers
151
views
Definition of conditional expectation for singleton
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-algebra. Furthermore, let $X, Y$ be two random variables from our probability ...
- 33
0
votes
0
answers
83
views
A closed form of mean-field equations
Assume that a system at time t, for example number of costumers in a line at time $t$ which is denoted by $q(t)$, follows a Markov chain with these dynamics (probabilities)
$$P(q(t+\Delta t)-q(t)=1)=\...
user135936
0
votes
1
answer
324
views
Finding examples of functions which are infinite or undefined with current extensions of the expected value?
Preliminaries
Consider the expectations desribed in this paper, which is an extension of the Lebesgue density theorem; this paper which is an extension of the Hausdorff measure, using Hyperbolic ...
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-1
votes
1
answer
312
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expectation of upper quantile proportion
(edited considerably following comments)
We have a collection $\boldsymbol{S}$ of $n$ discrete random variables $X_1$, $X_2$, $\dots$, $X_n$ $\overset{\small \text{i.i.d.}}{\small \sim}$ $\mathcal{D}$...
- 121
-2
votes
1
answer
118
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^...
