All Questions
Tagged with pr.probability probability-distributions
1,384 questions
4
votes
1
answer
134
views
The mean value of the reconstruction complexity of a random sequence
This problem is motivated by the problem of reconstructing a genome from the family of its short subwords.
Given a word $w$ and a positive integer $k$, let $M_k(w)$ be the family of all subwords of ...
- 41.9k
1
vote
1
answer
266
views
Decomposition of the sum of nonnegative random variables [closed]
Non-necessarily independent random variables $X_1,~X_2,~\cdots,~X_n$ are supported on $[0,a_1],~[0,a_2],~\cdots,[0,a_n]$ and with mean values $\mu_1,~\cdots,~\mu_n$ respectively, where all $a_i$ and $\...
- 550
8
votes
1
answer
2k
views
General Fourier inversion formula (Gil-Pelaez)
Gil-Pelaez (1951) proves the Fourier inversion formula
\begin{align*}
F(x) &= \frac{1}{2} + \frac{1}{2\pi} \int_0^\infty \frac{e^{itx}\phi(-t)-e^{-itx}\phi(t)}{it}dt \\
&= \frac{1}{2} - \frac{...
- 255
1
vote
1
answer
337
views
Posterior expected value for squared Fourier coefficients of random Boolean function
Let $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ be a Boolean function. Let the Fourier coefficients of this function be given by
$$ \hat f(z) = \frac{1}{2^{n}} \sum_{x \in \{0, 1\}^{n}} f(x)(-1)^{x \cdot ...
4
votes
0
answers
160
views
Can we show equivalence of two distributions based on their statistics?
Let $p,q$ be two distributions on $\mathbb{R}^d$. Let $f:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}$. Under what conditions does $\mathbb{E}_{x\sim p}f(x,z)=\mathbb{E}_{x\sim q}f(x,z)\ \...
- 57
0
votes
1
answer
496
views
Laplace transform inversion
I have a probability distribution that is defined through it's Laplace transform by :
$$L(t) = \mathbb E(e^{-tX}) = e^{1 - \frac{1+t}{t}\ln(1+t)}$$
Using R and the invLT package, i have a numerical ...
- 686
2
votes
1
answer
284
views
A distribution such that these expectation are 'closed-form'
I am seeking a continuous distribution with real positive support for the random variable $X$ such that, for all $t \in \mathbb R_{+}$,
$$\mathbb E \left(\ln\left(1+tX\right)\right)$$ is given in a '...
- 686
2
votes
0
answers
84
views
approximate the square of 2-norm distance between binary distributions with high probability
Suppsose we take $m$ samples from a Bernoulli distribution with probability $p$, and $m$ samples from another probability distribution with probability $q$. We want to calculate a statistic $x$ from ...
- 1,503
1
vote
0
answers
158
views
Is there a name for a random variable that is the absolute value of the difference between two iid discrete uniform variables?
I'm working on a project and I needed to calculate the distribution of the difference between two iid discrete uniform variables (sorry for the long title).
That is, let $I, J$ be two iid discrete ...
- 27
3
votes
1
answer
395
views
Symmetric distribution optimization problem of distances between points in $[0,1]$
Let $\mathcal{D}$ be a probability distribution with support $[0,1]$. Let $x, y, z$ the outcomes of three i.i.d. random variables $X, Y, Z$ with distribution $\mathcal{D}$, sorted in increasing order, ...
- 1,677
0
votes
0
answers
92
views
Linear independence of Wishart matrices
Let $W\sim W_n(I,d)$ be a real Wishart matrix of an identity covariance matrix and $d$ degrees of freedom, i.e., $W=XX^T$ for $X$ being an $n\times d$ matrix whose entries are i.i.d sampled from a ...
- 123
2
votes
1
answer
218
views
Probability distribution optimization problem of distances between points in $[0,1]$
Let $\mathcal{D}$ be a probability distribution with support $[0,1]$. Let $X, Y, Z$ three i.i.d. random variables with distribution $\mathcal{D}$, and $T$ a random variable uniformly distributed in $[...
- 1,677
0
votes
0
answers
173
views
The reason why a test is undersized?
Now I have a statistic $T_n$ for testing $H_0 \leftrightarrow H_1$, and I have proved that:
$$n T_n \rightarrow_d \chi_K^2$$
under $H_0$. Then an asymptotic $\chi^2$ test can be used, an asymptotic ...
- 331
1
vote
0
answers
46
views
How to use the mixed normal distribution to construct a proper statistics?
For a random vector $\xi_n \in \mathbb{R}^p$, if $\xi_n \rightarrow_d N(\mu, \Sigma)$, we can construct
\begin{equation*}
\Psi := \xi_n^{\top} \widehat{\Sigma}^{-1} \xi_n
\end{equation*}
for ...
- 331
1
vote
1
answer
2k
views
Convolution of two Gaussian mixture model
Suppose I have two independent random variables $X$, $Y$, each modeled by the Gaussian mixture model (GMM). That is,
$$
f(x)=\sum _{k=1}^K \pi _k \mathcal{N}\left(x|\mu _k,\sigma _k\right)
$$
$$
g(y)=\...
- 41
0
votes
1
answer
304
views
Anomaly with a pdf
The pdf of the range $\omega$ of $n$ identically r.v.'s random variables distributed with cdf $\mathbf{F}$ and pdf $\mathbf{f}$ is given by
$$ g(\omega)=n(n-1) \int_{-\infty}^{\infty} f(x)[F(...
- 826
2
votes
1
answer
905
views
Diagonalizability of Gaussian random matrices
Let $X$ be an $n\times n$ matrix whose elements are i.i.d. sampled from a normal distribution of zero mean and unit variance. Is $X$ diagonalizable over $\mathbb{C}$ with probability 1? Is there a ...
- 123
1
vote
0
answers
33
views
Condtions for a stochastic process to be locally non-factorizable
Given a stochastic process $X=(X_t)_{t\in I}$ on $\mathbb{R}^d$ with continuous sample paths supported on a prob. space $(\Omega, \mathscr{F}, \mathbb{P})$ and such that each pair $(X_s, X_t)$, with $(...
- 463
2
votes
0
answers
243
views
An inequality of KL Divergence for two different distributions passing through a same channel
Let $X$ be a random variable which takes values in $\mathcal{X}$. Assume that we pass $X$ through two independent conditional pdf $p_{X_1|X}$ and $p_{X_2|X}$ and choose $X_1$ with probability $\lambda$...
- 287
1
vote
2
answers
113
views
If a joint density factorizes on a square, does this imply that the marginal random variables are locally independent?
Let $Z=(X,Y) : \Omega\rightarrow\mathbb{R}^2$ be a Borel-measurable random vector and $U\subset\mathbb{R}$ be open. Suppose that $Z$ is absolutely continuous with Lebesgue density $\zeta$.
I was ...
- 463
1
vote
1
answer
130
views
How much reduction in expected variance can we get from a single bit?
Consider the following protocol:
Alice has a number $X$, chosen according to a known distribution $\mathcal D$ (e.g., $X\sim U[0,1]$).
She can send a bit to Bob, giving him more information about $X$ (...
- 127
2
votes
1
answer
160
views
A limit question involving Cramer's decomposition of normal random variables
I've come across the following question. Say we have two families of random variables, $X_N$ and $Y_N$, such that $\mathbb{E} X_N=\mathbb{E} Y_N=0$ and $\mathbb{E}X_N^2=1$. Now assume that:
$$|\mathbb{...
- 841
0
votes
1
answer
808
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
1
answer
149
views
Asymptotics of $\chi_m$-distribution where the degree of freedom $m \to \infty?$
I'm interested to see a result where for large degree of freedom $m,$ the chi distribution $\chi_m$ is increasingly well approximated by a family of normal distributions with parameters depending on $...
- 1,467
2
votes
1
answer
266
views
A random variable whose characteristic function decreases the fastest
A random variable $X$ is "good" for $(a_0, b_0) \in (0,1)^2$ if its characteristic function $\varphi_X(t)$ satisfies the following constraints:
$\forall t : \varphi_X(t) \geq 0$.
$\varphi_X$...
- 53
4
votes
1
answer
365
views
Reference for multivariate generalised CLT
I know that one can generalise the classical CLT in terms of heavy tail distributions, namely, for any i.i.d. random variables $X_i$,
$$\frac{X_1+\cdots+X_n}{n^{1/\alpha}}\rightarrow S(\alpha,\beta,\...
- 380
3
votes
3
answers
2k
views
Probability of a given string being a substring of another string
I am in interested into the following problem. We are given an alphabet $\Sigma$ of $k$ letters and a fixed string $S_1$ of length $l$ defined over $\Sigma$. Given a probability distribution $D$ over $...
- 41
0
votes
1
answer
428
views
First and last order statistics and their ratio for $\chi^2_{m}$ random samples
Let $X_1, \dots, X_n \sim_{iid} \chi^2_{m}$ be a random sample from a chi-squared distribution with $m$ degrees of freedom (d.f.). I was wondering if there's any known result for the order statistics
$...
- 1,467
4
votes
1
answer
1k
views
Functional derivative of differential entropy
I have trouble finding the derivative of the differential entropy w.r.t the probability density function, i.e. what is $\frac{\delta F[p]}{\delta p(x)}$, where $F[p] = \int_X p(x)\ln(p(x))dx$, and $p(...
- 41
2
votes
1
answer
1k
views
Sum of indicator functions of binomial random variables
Let $x_1, x_2,..., x_m$ be iid binomial random variables (each with a number of trials n and probability of success in each trial p). Define a list of binary indicator variables $y_1,y_2,...,y_m$ for ...
- 105
10
votes
3
answers
803
views
Discrete entropy of the integer part of a random variable
Let $X$ be a real valued random variable. Of course, the integer part $\lfloor X \rfloor$ of $X$ is a discrete random variable taking values in $\mathbb{Z}$. We can therefore define its discrete ...
- 2,306
0
votes
2
answers
174
views
Asymptotic properties of ANOVA when the number of groups goes to infinity
Suppose
$$X_{ij} = \mu_j + \varepsilon_{ij}, \quad j = 1, \cdots, J, \quad i = 1, \cdots, N_j$$
ANOVA can allow us to test whether $\mu_1 = \cdots = \mu_J$.
In traditional ANOVA, however, the number ...
- 331
1
vote
0
answers
212
views
A new notion of probability coupling
Let $X$ and $Y$ be two discrete random variables distributed according to $\mu$ and $\nu$, respectively. Consider the following optimization problems
$$\inf_{\pi\in \Pi(\mu, \nu)}\Pr(X\neq Y),$$
...
- 1,109
2
votes
1
answer
181
views
Conditional entropy - solve example
Given a random variable $X$ that is uniformly distributed on $[-b,b]$ and $Y=g(X)$ with
$$g(x) = \begin{cases} 0, ~~~ x\in [-c,c] \\ x, ~~~ \text{else}\end{cases}$$
Now I want to compute the ...
- 131
4
votes
1
answer
320
views
The power of chi-square test
Under the null hypothesis, if we have
$$\sqrt{n} \vec{x} \, \rightarrow_d \, N(0, I_p),$$
the test statistic can be construct as:
$$\hat{\Psi} = n \vec{x}^{\top} \vec{x} \, \rightarrow_d \,\chi^2_p.$$
...
- 331
1
vote
2
answers
139
views
Inaccurate results for the analytical expression of $\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]$
I'm trying to plot a graph for the following expectation
$$\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
1
answer
220
views
Distributions associated with random sets and sums of random sets
Let's say you have an infinite random set $S$ of non-negative integers, and $T=S+S=\{x+y$ with $x,y\in S\}$. Let $N_S(z)$ be the number of elements of $S$ less than or equal to $z$; it is a random ...
- 3,098
0
votes
0
answers
93
views
Regularity with respect to the Lebesgue measure through dimensions
Let us consider two probability measures $\mu \in \mathcal{P}(\mathbb{R}^{p})$ and $\nu \in \mathcal{P}(\mathbb{R}^{q})$ with $p,q \in \mathbb{N}^{*}$. We note $\#$ the push forward operator i.e for $...
- 407
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 ...
2
votes
1
answer
300
views
On the speed of divergence of the converse of the Strong law of large numbers
By the converse of the strong law of large numbers, we know that, given a sequence of i.i.d random variables $X_1,X_2,\dots$ such that $\mathbb{P}(X_1 \ge 0)=1$ and $\mathbb{E}X_1= \infty$,
then I ...
- 446
1
vote
0
answers
62
views
Quantitative bounds on convergence of Bayesian posterior
Let $Y$ be a random variable in $[0,1]$, and let $X_1, X_2, \ldots$ be a sequence of random variables in $[0,1]$. Suppose that the $X_i$'s are conditionally i.i.d given $Y$ ; in other words, I'd like ...
- 173
3
votes
1
answer
984
views
About the metrizability of the space of Probability measures $\mathcal{P}(S)$
It is often proved in Books that the space of Probability measures $\mathcal{P}(S)$ on a Polish metric space $(S,\rho)$ endowed with the weak/narrow topology induced by declaring it to be be the ...
- 309
1
vote
1
answer
169
views
Probability involving dependent random variables constructed from i.i.d. Gaussians
This is a problem I need to address for a certain computation in my research.
Let $Y_1,\dots,Y_n$ be a sequence of i.i.d. standard normal variables; and let $I\subset[0,+\infty)$ be an interval. In my ...
- 1,096
2
votes
1
answer
201
views
Continuous version of conditional probability distributions $( \mathcal{L}(X_t | \mathcal{G}) )_{t \geq 0}$ if $(X_t)_{t \geq 0}$ is continuous?
Let me first explain the setup:
Let $(X_t)_{t \geq 0}$ be a stochastic process on some probability space $(\Omega,\mathcal{F},P)$ with values in a complete and separable metric space $E$ (e.g. $E = \...
- 309
5
votes
2
answers
2k
views
Tight sequence of measures
This is probably a very easy question for experts in probability or measure theory.
I have a sequence of finite measures $\mu_{n}$ on a non-compact metric space $X$ such that $\mu_{n}$ converges to $\...
- 38
12
votes
1
answer
628
views
A function with unexpectedly simple Legendre transformation
Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and
\begin{equation}
I(x)=
\begin{cases}
\frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\
\...
- 1,608
30
votes
8
answers
3k
views
A variation of the law of large numbers for random points in a square
I uniformly mark $n^2$ points in $[0,1]^2$. Then I want to draw $cn$ vertical lines and $cn$ horizontal lines such that in each small rectangle there is at most one marked point. Surely, for a given ...
- 5,063
2
votes
0
answers
120
views
Average of random variables is "more log-concave"
Problem (1)
Suppose $\phi_i\in [0,\pi/2]$ are drawn uniformly for $1\le i\le n$, and $\sum_{i=1}^n w_i=1$, $w_i\ge 0$. Show that the pdf $p_1$ of the random variable
$$\phi = \sin^{-1}\left(\sqrt{\...
- 371
3
votes
1
answer
335
views
Reversing the order of conditioning in a sum to compare conditional variances
Suppose $Z=X+Y$ where $X$ is independent of $Y$ and $Y\sim N(0,1)$. I would like to compare $\text{var}(E(X|Z))$ to $\text{var}(E(Z|X))$. Obviously, $\text{var}(E(Z|X))=\text{var}(X)$.
In particular, ...
1
vote
2
answers
369
views
How to solve this stochastic optimization problem?
How one can solve the following stochastic optimization problem?
\begin{align}
\max\quad& \mathbb{E}[\mathbf{1}^{\mathrm{T}}X]\\
\text{s.t.} \quad& \mathbb{E}[\mathbf{A}X]\leq\mathbf{1}_{m\...
- 287