**0**

votes

**0**answers

8 views

### How to obtain a unimodal histogram with normal distribution (gaussian)? [on hold]

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**

votes

**1**answer

52 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**

votes

**2**answers

94 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/...

**1**

vote

**0**answers

54 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")...

**0**

votes

**0**answers

25 views

### Application of Lemma in Iterated Expectation [on hold]

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 \...

**0**

votes

**0**answers

37 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 ...

**0**

votes

**0**answers

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 ...

**1**

vote

**1**answer

42 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**

votes

**0**answers

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**

votes

**1**answer

133 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**

votes

**1**answer

80 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**

votes

**2**answers

237 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**

votes

**0**answers

55 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
$$
...

**0**

votes

**0**answers

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**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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 ...

**21**

votes

**3**answers

711 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 $n$?
...

**1**

vote

**0**answers

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 ...

**1**

vote

**2**answers

59 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 ...

**1**

vote

**0**answers

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 ...

**0**

votes

**0**answers

33 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}...

**0**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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**

votes

**0**answers

73 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}^{-...

**1**

vote

**0**answers

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**

vote

**1**answer

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 ...

**1**

vote

**1**answer

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**

votes

**1**answer

148 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 ...

**1**

vote

**1**answer

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 $\sum_1^n k Z_k = n$:
$$\...

**1**

vote

**1**answer

107 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 ...

**0**

votes

**0**answers

30 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: $...

**1**

vote

**0**answers

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,....,...

**0**

votes

**0**answers

36 views

### 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**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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 $\mathscr{P}_1$ (...

**1**

vote

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

136 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 ...

**1**

vote

**0**answers

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**

votes

**1**answer

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,\...

**1**

vote

**0**answers

30 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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**2**answers

150 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 $...

**1**

vote

**0**answers

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 ...

**1**

vote

**1**answer

86 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**

votes

**1**answer

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 $f$. ...

**2**

votes

**0**answers

44 views

### A question about probabilistic graphical models

Say one is given a probabilistic graphical model and a cut of the underlying graph. Do we know any statements about when and how can one or many of the marginals (of the sources) or the conditionals (...

**2**

votes

**0**answers

89 views

### Growth of inner products between two random vectors on the sparse hypercube

We define the $s$-sparse hypercube in $\mathbb{R}^d$ as
\begin{align}
\mathbb{H}_s = \bigl \{ {\bf{v}} \in \{ -1, 0 , 1\}^d \colon \| {\bf{v}} \|_0 = s \bigr\},
\end{align}
where $ \| {\bf v} \|_0 $ ...