Questions tagged [expectation]
The expectation tag has no usage guidance.
1
vote
0answers
19 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 '...
4
votes
1answer
116 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
54 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
107 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). ...
4
votes
1answer
253 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
$$
\mathbb{E}_{ZZ'}\left[ \exp\...
0
votes
0answers
48 views
Upper bound on expectation of product
I want to upper-bound the following quantity:
$$\mathbb{E}_Y\left[f(Y)g(Y)\right] $$
The idea would be to get something of the shape: $\mathbb{E}_Y[f(Y)]\cdot h(Y)$
where $h(Y)= j(\mathbb{E}_Y[k(g(Y))]...
0
votes
0answers
69 views
Expected value of eigenvalue of matrix
Let $A = (X_{ij})_{ij}$ a square matrix of size $n$ where the $X_{ij}$ are (discrete) real random non-negative entries. Denote by $\lambda_1(A) \geq \dots \geq \lambda_n(A)$ the (random) ordered ...
1
vote
0answers
27 views
Expected value of inverse of complex non-central Wishart matrix
I have a matrix $W$ that abides a complex non-central Wishart distribution.
My question is what the expectation of the inverse is, i.e., how to compute
$$\mathbb{E}(W^{-1}).$$
I have tried to read up ...
1
vote
2answers
118 views
lower bound the probability of at least L collisions
Lets say we get a list $M$ containing $|M|=\sqrt{L\cdot N}$ randomly and independtly drawn elements from a set of size $N$. And lets denote the $i$-th element of the list $M$ by $M[i]$.
If we now ask ...
1
vote
1answer
55 views
On the eigenvalue of the expectation value of a random matrix in quadratic form
When we handle with some dynamic input-output mappings, there occurs a question as follows:
Let $M$ be a random matrix, of which each element contains random terms. Consider the two expectation ...
1
vote
0answers
26 views
Mean and correlation of product of two random processes
I have two random process:
$$A(at)$$
$$\cos(2\pi f_0t+\Phi)$$ with these hypothesis:
$a$ and $f_0$ are constant
$\Phi$ is uniformly distributed in $[0,\pi)$
$A(at)$ is WSS
I must calculate the ...
1
vote
0answers
59 views
If $f$ is a measurable random field, then $(ω,x)↦E[f(x)\mid F](ω)$ has a measurable version $g$ and $E[f(X)\mid F]=g(X)$ for all $F$-measurable $X$
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$\mathcal F\subseteq\mathcal A$ be a $\sigma$-algebra on $(\Omega,\mathcal A)$
$(E,\mathcal E)$ be a measurable space
$f:\Omega\times ...
3
votes
1answer
220 views
Approximating the expectation of a matrix inverse
Let
$$R := A \Lambda^{-1} A^H + \frac{1}{\gamma} I_n$$
where $A$ is a given $n \times m$ matrix (where $m \gg n$),
$$\Lambda := \mbox{diag} \big( \lambda_1, \lambda_2, \dots, \lambda_m \big)$$
...
8
votes
1answer
221 views
Expectation inequality for sampling without replacement
Is the following proposition correct?
$X_1, X_2, X_3$ are uniformly at random sampled from a finite set $\mathcal X$ without replacement.
$f : \mathcal X^2 \rightarrow \mathbb R_{\ge0}$ is symmetric:...
3
votes
1answer
236 views
Expected value of the maximum of the periodogram
Let us suppose that $X_1,\ldots,X_n$ with $n\ge1$ are iid random variables such that $\operatorname EX_1=0$ and $\operatorname E|X_1|^s<\infty$ with some $s>2$ and define the DFT of $X_1,\ldots,...
