All Questions
Tagged with expectation probability-distributions
34 questions
1
vote
1
answer
53
views
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
2
votes
1
answer
73
views
Connection between Wassertein-2 metric and difference in variance
Given two probability densities $\mu\in\mathcal P(\mathbb R^d)$ and $\nu\in\mathcal P(\mathbb R^d)$, we define their Wasserstein-$p$ metric as
$$
W_p^p(\mu, \nu)=\inf_{\gamma\in \Gamma(\mu, \nu)}\int_{...
- 123
1
vote
1
answer
111
views
Expected number of solutions of a random quadratic polynomial system over a finite field
Let $\mathbb{F}_q$ be a field of $q$ elements. Let $a_{i,j,k}$, $b_{i,j}$, $c_i$ ($1 \leq i \leq m$, $1 \leq j \leq k \leq n$) be independent uniformly distributed random variables in $\mathbb{F}_q$, ...
- 157
14
votes
1
answer
1k
views
A disc contains many random points. Each point is connected to its nearest neighbor. What is the expectation of average cluster size?
A disc contains $n$ independent uniformly distributed points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points.
For example, here are $20$ random ...
- 3,507
0
votes
2
answers
239
views
Computing the expectation of a quadratic matrix form involving Bernoulli and Gaussian distributed matrices
I am working with two random matrices, $Z$ and $H$:
$Z$ is an $n \times K$ matrix with entries sampled i.i.d. from a Bernoulli distribution: $Z_{ij} \sim \mathrm{Bernoulli}(p)$.
$H$ is a $K \times K$ ...
- 37
0
votes
0
answers
46
views
Prove lower collinearity on the tails of Gaussian blob
Let us consider a $n$-dimensional Gaussian blob, i.e. a set of $N$ random vectors $\{\boldsymbol{X}^{(j)}\}_{j=1}^N$, with $n$ independent components, $X_i^{(j)}$, and such that $X_i^{(j)} \sim \...
- 305
1
vote
1
answer
204
views
Expected (maximum minus minimum) of Laplacian random variables
Suppose there are $n$ IID random variables denoted as $X=(X_1,\dots, X_n)$, they follow Laplace distribution with parameter $\lambda$, denoted as $Lap(\lambda)$. That is,
$$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
31
votes
5
answers
2k
views
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 ...
- 3,507
1
vote
1
answer
264
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 =...
- 43
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
views
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 ...
0
votes
1
answer
390
views
expectation of log(1-x^a) if x is a beta random variable
How can I compute $\mathbb{E}_{q}\Big[\log (1-x^a)\Big]$ when the distribution of $q$ is given as $q(x)\sim\mathrm{Beta}(\alpha,\beta)$?
- 37
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 ...
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
9
votes
1
answer
482
views
Bounds on the expectation of $|X-Y|$ for $X,Y$ Poisson
I would have a proof of the following fact; but it's a bit clunky, and am wondering if one can get a more elegant one (and/or improve the constants). I couldn't find this anywhere, and searching ...
- 1,372
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
0
votes
1
answer
387
views
Simplification on the estimation on error of the ratio of 2 random variables
Let $Z=\dfrac{X}{Y}$ the ratio of 2 random variables.
Distribution of $Z=\dfrac{X}{Y}$
Consider the case of two independent normal variables $X$ and $Y$ with strictly positive means and variances $\...
0
votes
1
answer
802
views
Concentration of $\ell_2$ norm of a vector sampled from a distribution
Let $X=(X_1,\ldots,X_n)$, where $X_i \sim P_{p_i}(0,\frac{1}{\lambda})$ are iid, $P_{p_i}$ is sub gaussian distribution for $i^\text{th}$ element, and 0 and $1/\lambda$ are mean and variance.
I'm ...
- 61
1
vote
2
answers
139
views
Inaccurate results for the analytical expression of $\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]$
I'm trying to plot a graph for the following expectation
$$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]=a 2^{-\frac{\kappa }{2}-1} b^{-\frac{\kappa }{2}} \theta ^{-\kappa } \...
0
votes
2
answers
246
views
Finding the expectation of $a \mathcal{Q} \left( \sqrt{b } \gamma \right) $, where $\gamma$ is a Gamma r.v
I'm trying to analytically find the following expectation
$$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right],$$
where $a$ and $b$ are constant values, $\mathcal{Q}$ is the ...
4
votes
2
answers
852
views
Disintegration, conditional probabilities, and conditional expectation
On the Wikipedia page there is a note that conditional probability measures can be described by disintegration. However, I can seem to find a clear exposée of how this construction is related to ...
- 5,407
1
vote
1
answer
139
views
Conditional density for random effects prediction in GLMM
I am currently working on generalized linear mixed models (GLMM) and need some help concerning the prediction of the random effects. More specifically, I don't understand the given representation of ...
- 13
0
votes
1
answer
181
views
expectation of a quadratic function of a matrix variate normal distribution
I want to compute the following expectation term:
$E[{\bf{XA}}{{\bf{X}}^T}]$
where ${\bf X} \in R^{M \times M}$ and its elements are normal random variables such that
$vec\left( {\bf{X}} \right)\...
- 275
0
votes
1
answer
295
views
Are there known bounds on the expectation of the truncated Beta distribution?
Let $X\sim beta(\alpha,\beta)$ be a random variable and let $\tau\in(0,1)$.
Are there any known closed-form bounds (I'm specifically interested in lower bounds) on
$$
\mathbb E[X\ | X\le \tau]?
$$
- 618
1
vote
1
answer
126
views
The expectation of binary logistics regression with respect to Gaussian distribution
I am trying to compute the expectation of $g(s,x)=s \ln \sigma(x)+(1-s)\ln(1-\sigma(x))$ with respect to the normal distribution $\mathcal{N}(x;m,v)$, where we have $\sigma(x)=\frac{1}{1+e^{-x}}$. If ...
- 37
3
votes
1
answer
3k
views
Is there a tight lower bound for the expectation of the product of two positive valued random variables?
Let $X,Y$ be two (dependent) random variables with $\mathbb{P}(X\ge 0)=\mathbb{P}(Y\ge 0)=1$.
I want to find a tight lower bound of $\mathbb{E}(XY)$ when $X,Y$ are non-negative, almost surely.
...
23
votes
5
answers
2k
views
Maximizing the expectation of a polynomial function of iid random variables
Let $f \colon \mathbb R^N \to \mathbb R$ be a smooth function. Let $\mu$ be a probability measure on $[0,1]$ and $X_1, \ldots , X_N$ be i.i.d. random variables on $\mathbb R$.
Question 1. What is ...
- 980
3
votes
2
answers
259
views
What is $\sum_{k=0}^{+\infty}{k⋅p(k;\mu_1,\mu_2)}$, where $p$ is the pmf of Skellam distribution?
The Skellam distribution is the discrete probability distribution of the difference $N_{1}-N_{2}$ of two statistically independent random variables $N_{1}$ and $N_{2}$, each Poisson-distributed with ...
- 181
1
vote
0
answers
269
views
Expected value of inverse of complex non-central Wishart matrix
I have a matrix $W$ that abides a complex non-central Wishart distribution.
My question is what the expectation of the inverse is, i.e., how to compute
$$\mathbb{E}(W^{-1}).$$
I have tried to read up ...
- 11
1
vote
2
answers
461
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 ...
- 123