In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.

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Comparing log functions of CDFs and PDFs (related to order statistics) with non-log functions of the same

Let $f$ and $F$ denote the respective pdf and cdf of a probability distribution on $\mathbb{R}$. Consider any natural $n\geq3$ and any real $a$ and $c$ such that $a\leq c$, and $\rho\geq0$. We want ...
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79 views

Proving that an integral related to order statistics is increasing in a certain parameter

Let $f$ and $F$ denote, respectively, the pdf and cdf of a probability distribution on $\mathbb R$. Take any natural $n\ge3$ and any real $a$ and $c$ such that $a\le c$. Does it always follow that $$...
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8 views

How to obtain a unimodal histogram with normal distribution (gaussian)? [closed]

My task is to come up with a histogram consisting of $N$ bins. The histogram should show a (perfect) normal distribution. So something similar to what is shown in this image. How do I obtain the value ...
2
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1answer
54 views

How to compute bounding coefficients for McDiarmid's inequality?

I am trying to understand the proof in Sec. A2 of Gretton et al.. To make the question self-contained, I summarize below the key ingredients. At the end of the post, I state my question. Given a ...
6
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2answers
98 views

Uniform Concentration Bounds on Weighted Sum of i.i.d. Bernoulli Random Variables

Let $\delta_1,...,\delta_n$ be $n$ independent identically distributed Bernoulli random variables with $\mathbb{P}(\delta_1=1)=p$. We consider a set $\Omega = \{\mathbf{a}:=(a_1,...,a_n)~|~a_i\in [0,c/...
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1answer
78 views

Reference on Probability theory on functional spaces (in special Hilbert spaces)

Currently, I am working on some sort of stochastic optimization problems defined over function spaces. I am familiar with standard probability theory (R. Durrett, ''Probability: Theory and Examples")...
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25 views

Application of Lemma in Iterated Expectation [closed]

I was reading the following paper: InfoGAN. I cannot figure out, how on page 4, Lemma 5.1 was applied in the following lines: $$\mathbb{E}_{c \sim P(c), x \sim G(z,c)}[\log Q(c|x)] = \mathbb{E}_{x \...
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40 views

Last Inference in proof of conditional limit theorem

I read about the Conditional Limit Theorem from the book "Elements of Information Theory" by Thomas M. Cover and Joy A. Thomas, second edition, page 371. I can't understand the last inference in the ...
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21 views

Article Using Kullback Leibler Divergence to Measure Divergence of Observation from Distribution

I am currently attempting to compare an observed distribution to a theoretical distribution, and my current approach is to normalize the two and find the Kullback Leibler Divergence. I am beginning to ...
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1answer
43 views

Difference between Pareto-Levy and Pareto distributions

Many authors use the term Pareto-Levy distribution, though Im not clear how these are different from Pareto. Are these also Power Law distributions and is there a way of visually confirming if an ...
5
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42 views

Why is it easy to compute the first and fourth moments for random chord length in a convex solid?

Recently I was led to some considerations in geometric probability, a field pretty far from any specialization of mine. (Context: I was working with a collaborator on a question about mean escape ...
4
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1answer
134 views

Uniform sampling from general simplex with a twist

This is part of a question I had asked elsewhere, and then some of the links redirected me to CS stack exchange. Given $0\leq a_1\leq\dots\leq a_D\leq1$ (all strictly positive), I want to draw points ...
3
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1answer
83 views

Probability of collision of some family of hash functions

Given $x$ and $y$ in $\mathbb{R}$, and let $\mathcal{H} = \{ h \mid \mathbb{R} \to \mathbb{N} \}$ be a family of hash functions where $ h(x) = \left\lfloor x + \sum^C_{i=1} U_i \right\rfloor$ for some ...
4
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2answers
239 views

Negative probabilities - what are two ordinary pgfs that correspond to the gf of a half-coin?

In Half of a Coin: Negative Probabilities, author considers pgf of a fair coin represented by random variable, $X = 1_H$: $$G_X(z) = E[z^X] = \sum_{x=0,1} z^xP(X=x) = (z^0)(1/2) + (z^1)(1/2) = \frac{...
7
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57 views

Distribution of the stopping time of an autoregressive sequence

Consider $e_t$ be i.i.d. uniformly chosen from $\pm 1$. Let $\eta$ be a small positive constant. What is the distribution of $T$ such that $\eta^{0.5} (1+\eta)^T W_T$ first hits $\pm 1$, in which $$ ...
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42 views

What is the inverse of the integrated $\chi^2$ function?

I am implementing some preprocessing of variables in the context of a paper called A Neural Bayesian Estimator for Conditional Probability Densities. It states: 1.) Given a non-linear, a monotonous ...
3
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1answer
77 views

Tail bound for product of normal distribution

Let $U, V$ be two standard normal random variables with covariance $cov(U,V) = \beta \in [0,1)$. Let $W = UV$ be the product of two RV's, and $W_1, W_2, \ldots, W_n$ be n i.i.d copies of $W$, what's ...
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76 views

Looking for an exposition of a certain theorem of Talagrand

The following is a theorem by Talagrand (as stated here, http://arxiv.org/pdf/1511.08609v1.pdf), Let $(X, \mu)$ be a probability space. Let $F : X \rightarrow \{0,1\}$ be a family of functions ...
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3answers
714 views

On the sum of uniform independent random variables

Let $X_1,...,X_n$ be independent uniform random variables in [0,1] and assume $c>1/2$. Is it true that $$\mathbb{P}\left[\sum_{i=1}^n X_i \leq n \cdot c\right]$$ is increasing with respect to $n$? ...
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67 views

How to prove this Gaussian Mixture theorem? (Fitting/Overfitting)

Note from OP: I gave up and reposted this Question with a Bounty on Cross Validated HERE. In certain applications, we approximate an unknown pdf by placing uniformly weighted Gaussian terms at each ...
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2answers
61 views

Reference for the monotonicity in $\alpha$ of the Rényi entropy

I'd like to have a reference for the property $0 \leq \alpha < \alpha' \leq \infty \implies R_\alpha(\mu) > R_{\alpha'}(\mu)$, where $R_\alpha(\mu)$ is the Rényi entropy of order $\alpha$ of a ...
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57 views

Bounding a distribution using moments

Suppose $X$ is a non-negative random variable with bounded image. I was wondering if anybody knew of any results that could answer a question of the following type: Suppose the $n$-th moment satisfies ...
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35 views

information about composite random process

I have a following composite random process $$X_j = v_0 + 1/j^2 + Y_j + Z_j$$ where $v_0$ is a constant, $Y_j \rightarrow 0$ almost surely as $j\rightarrow \infty$ and $Z_j \sim N\big(0, \frac{a^{2j}...
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1answer
56 views

Discretization of a continuous distribution

For a research project I work with continuous distributions, like the normal distribution. In my use case however the random variable Z generally follows a normal distribution, though it can only take ...
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99 views

Radon-Nikodym for continuous time processes

Likelihood theory for statistical inference concerning stochastic processes in continuous time are well used. How ever i've found no real literature concerning the fundamentals. What is know from ...
5
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77 views

Distribution of Random Knots from Braids

Let $R_{2n,l}$ be a random braid word of length $l$, where each letter is chosen uniformly from the braid generators of $B_{2n}$, $\{\sigma_1,\ldots,\sigma_{2n-1},\sigma_1^{-1},\ldots,\sigma_{2n-1}^{-...
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35 views

Expression for Joint-PDF of Langevin equation?

How to derive exact or approximate analytical expression for time-dependent joint-PDF (velocity-coordinate PDF) for Langevin equations of Brownian motion? Langevin equations is: $\dot{x}=v$ $\dot{...
1
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1answer
43 views

Maximizing joint entropy?

I'm stuck trying to find the maximum entropy probability distribution taking into account a joint distribution. Basically, I want to find the maximum entropy expression for $p(x,y)$ when the marginal ...
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1answer
48 views

A generalization of negative binomial distribution

Assume we have a set of n balls. For each step, we uniformly pick one ball and label it if it is not labeled. Or otherwise move on to next step. I am wondering what is the distribution of number of ...
2
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1answer
150 views

Ask for a special function related to the error function

I am wondering whether anyone knows the following integration has a named special function or a reference $$ F_{a,b}(z) :=\frac{2}{\sqrt{\pi}} \int_0^z \text{erf}(a+b y)\: e^{-y^2} \text{d}y $$ for ...
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1answer
49 views

Conditioned sum of n Poissons versus unconditioned Poissons

Let $\theta >1$ and take independent random variables $Z_k \sim \text{Poisson}(\theta/k)$ for $1 \leq k \leq n$ and let $Z_k^*$ have marginals like the $Z_k$ conditioned on $\sum_1^n k Z_k = n$: $$\...
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1answer
120 views

Convexity of truncated expectation

Let $k, n$ be two positive integers with $k \leq n$, and let $P = \{ (x_1, \dots, x_n) \in [0, 1]^n : \sum_i x_i = k \}$. Given $x = (x_1, x_2, \dots, x_n) \in P$, let $X_i$ be the random variable ...
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32 views

Product of lognormal random variables

Let $X_1$ and $X_2$ be two normal random variables. Write $X_1\sim N(\mu_1, \sigma^2_1)$ and $X_2\sim N(\mu_2, \sigma^2_2)$, to fix ideas. Consider the corresponding log-normal random variables: $...
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42 views

Characterize Linear Transformation of Dirichlet Distribution

Let $X=(X_1,....,X_K)\sim{}\text{Dir}(\alpha_1,...,\alpha_K)$ be a Dirichlet distribution with parameters $\alpha_1,...,\alpha_K$. Let $A$ be a non-singular linear map and $(Y_1,....,Y_K)=A(X_1,....,...
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Proving Fixed Point Algorithms

In Thomas Minka's paper on Estimating the Dirichlet Distribution (link here http://research.microsoft.com/en-us/um/people/minka/papers/dirichlet/minka-dirichlet.pdf), the author presents a fixed ...
2
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42 views

Existence of probability distribution satisfying upper/lower bounds on events

Suppose we have a finite sample space $S$ and some events $A_1, \dots, A_k \subseteq S$. We would like to put a probability distribution on $S$ so that no element has probability greater than a ...
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61 views

The role of absolute continuity in stochastic ordering defined over sets of probability distributions

This question is about a claim given in this paper (page 261, the remark), but without any proof. It simply says that if two sets of probability distributions, $\mathscr{P}_0$ and $\mathscr{P}_1$ (...
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35 views

On the numerical range of non-self adjoint Gaussian matrix

For a complex $n \times n$ matrix $A$, its numerical range is the set $$W(A) = \left\{\mathbf{x}^*A\mathbf{x} \mid \mathbf{x}\in\mathbb{C}^n,\ \|x\|_2=1\right\} .$$ We can further define the ...
4
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1answer
81 views

On the eigenvalues' distribution of random unitary

Fix an integer $d$, let $\mathbb{U}_d$ be the $d\times d$ unitary group. For any $U\in \mathbb{U}_d$, define $\Omega(U)$ be the length of the smallest arc containing all the eigenvalues of $U$ on the ...
2
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176 views

Expected value and variance of a stochastic process

I would like to ask if there is a way to find the expected value and the variance of the following process $$ dv_t=(a-be^{\alpha v_t})dt+\sigma dW_t, \quad v_t=v_0 $$ where $a\in (-\infty,+\infty), b&...
5
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1answer
139 views

Upper Bound for the Difference of Even Probability and Odd Probability in Hypergeometric Distribution

Let $X$ be a random variable following the hypergeometric distribution with parameters $N,K,n$, where \begin{equation} Pr(X=k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}. \end{equation} To ...
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63 views

Multimodal property of polynomial logistic distribution

Let $P(x)$ be a polynomial (of an odd degree $n$) strictly increasing on $(-\infty, +\infty).$ Then $F(x)=\displaystyle \frac{1}{1+\exp\{-P(x)\}}$ is a distribution function of a polynomial logistic ...
2
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1answer
67 views

Median of a uniform multinomial variable

Let $k\in\mathbb N^+$ be a positive integer. Consider a set of i.i.d. random variables $X_1,X_2,\ldots, X_n$, each of which is distributed uniformly over $\{1,2,\ldots,2k+1\}$. For $i\in \{1,2,\...
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31 views

Expected number of perfect matchings in bounded degree bipartite graphs

Consider collection $\mathcal C_{n,n,\Delta}$ of every $2n$ vertex balanced bipartite graph of average degree $\Delta$. What is the expected number of perfect matching a graph in $\mathcal C_{n,n,\...
5
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1answer
216 views

Estimate of incomplete binomial integral

Let $0\le k \le n$. Prove that $$ n\binom{n}{k}\int_{0}^{\frac{k}{n+1}}t^k(1-t)^{n-k}\,dt \le 1/2. $$ As far as I know 1) it is proved for $\frac{k}{n+1}\le 1/2$ and 2) not proved for $1/2 <\frac{...
3
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34 views

Joint distribution on order statistics and sample history

If samples $X_1, X_2, ... X_t$ are picked independently and identically from the discrete uniform distribution $[1,2, ..., P]$, what is the joint distribution of the last $k$ order statistics and last ...
5
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2answers
151 views

Expected number of changes in the sign of a rolling sum of independent normal variables

Imagine we define $Y(t+n)= X(t+1)+.....+X(t+n)$ where $X(i)$ is an independent normal (i.e. everyday we remove the starting observation and we add a new one). We have $n$ consecutive observations of $...
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51 views

BM hitting times with exponential killing process

Assume a BM in 3d domain (infinite) with a small absorbing subdomain (cube, sphere, ect), centered at point $p_s=(x_s,y_s,z_s)$ . BM starts at point $p_0=(x_0,y_0,z_0)$ and when it riches the ...
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1answer
87 views

Fundamental difference between Poisson Point Process and Binomial Point Process

What is the fundamental difference between Poisson Point Process and Binomial Point Process? I am evaluating a solution in a Binomial Point Process setup. If I want to evaluate that in a Poisson ...
7
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1answer
107 views

Choosing a sample based on where the density function is highest

Is there a name for the following process? Say I have an absolutely continuous probability density function $f$ with compact support, and I take $k$ independent samples $x_1,\dots,x_k$ from $f$. ...