All Questions
Tagged with pr.probability st.statistics
1,134 questions
2
votes
2
answers
739
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Multinomial transformation for matrices
Suppose we have a vector of probabilities $\mathbf{p}=(p_1,...,p_n)$, where $p_i>0$ for $i=1,...n$ and $\sum p_i=1$. Define new vector $\mathbf{r}=(r_1,...,r_{n-1})$ in a following way:
$r_i=\log(...
CommunityBot
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0
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216
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Hoeffding's lemma for unbounded r.v with bounded exponential map
Let $X$ be a real r.v with $E[e^{\lambda X}] < \infty $ for all $\lambda \in [-c,c]$.
Is it possible to get an Hoeffding's lemma like bound on $E[e^{\lambda(X-EX)}]$. That is, an upper bound: $$E[...
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1
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1
answer
133
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Independence of two random variable
Let $W$ and $S$ are two positive valued continuous random variable. Suppose
$g: [0,\infty)\rightarrow [0,\infty)$ is a convex function with a constraint that $g$ can't be of the form $g(x)=cx$, $c$ ...
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1
answer
3k
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An Inequality of KL Divergence
Given two probability distributions $P$ and $Q$ defined over a finite set $\mathcal{X}$, one can define the KL divergence between $P$ and $Q$ as
$$D(P||Q):=\sum_{x\in \mathcal{X}}P(x)\log\frac{P(x)}{...
- 128k
2
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1
answer
141
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Convergence of a test statistic
I'm reading a paper of Shao and Zhang:
Testing for Change Points in Time series.
In this paper they claim the following:
The are testing whether there is a change in the mean of a time series. So
$H_{...
- 128k
2
votes
1
answer
474
views
Large deviations for sums of random variables whose correlation function decays exponentially
Let $X_1, \ldots, X_N$ be a string random variables taking values $X_i \in [-1,1]$ and jointly distributed according $P(X_1, X_2, \ldots, X_{N-1}, X_N)$, which is invariant under cyclic permutations ...
1
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0
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70
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Bounds on product of CDF or Beta function
I have functions of the form
\begin{align}
I_i = \int_0^\infty F_0(x)^aF_1(x)^b(1-F_0(x))^c(1-F_1(x))^ddF_i(x),~~~~i = 0,1.
\end{align}
$F_0(x)$ and $F_1(x)$ are CDFs corresponding to the random ...
- 19.6k
2
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0
answers
71
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Asymptotic results for functions of order statistics
There are $n$ ($n \ge 3$) iid random variables $\{ {c_i}\} _{i = 1}^n$ on the interval $[\underline c,\bar c]$ ($\underline c>0$). The cdf $F(\cdot)$ and pdf $f(\cdot)$ are unkown to us, but we ...
- 19.6k
2
votes
2
answers
3k
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Proof of conditional copula relation to the marginal copulas
Hello
I am trying to derive the second equation displayed in section 7.1 (or p. 41) of this article or equation (6.3) of this book. I've seen this in many documents discussing conditional sampling ...
- 101
16
votes
1
answer
2k
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Normal approximation of tail probability in binomial distribution
My problem: From the Berry--Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B_n$ has the standardized binomial distribution and $\Phi$ ...
CommunityBot
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2
votes
7
answers
4k
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Estimating the mean of a truncated gaussian curve
Say I have a black box generating data samples, and I want to estimate the parameters of the black box from the samples.
The black box works like this: it has a parameter m (a real number), and to ...
CommunityBot
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1
vote
0
answers
533
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Finding an error estimation for the De Moivre–Laplace theorem with Stirling's formula
Context for my question: For one part of my thesis I try to find an upper bound for the error in the normal approximation of the binomial distribution following the standard proof of the De Moivre–...
CommunityBot
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1
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163
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$\int_0^t f(s)\,dB_s$ normally distributed, mean and variance
Suppose that $f(t)$ is a (non-random) continuous function on $[0, \infty)$. Let$$Z_t = \int_0^t f(s)\,dB_s.$$
How do I see that $Z_t$ is normally distributed?
What is the mean and variance?
I need ...
- 29.8k
1
vote
0
answers
42
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Adding weights to the Brier score
Fix $n > 0$, and consider the space $\cal P$ of probability functions defined over the Boolean closure of a fixed $\cal S = \{ s_1, \ldots, s_n \}$. The Brier score of $P \in \cal P$ at $s_i \in \...
- 111
1
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1
answer
256
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KL divergence Inequality
I am trying to find a proof for the following inequality, but I did not get anywhere following the references from the paper I was reading.
Consider two probability measures $P$ and $Q$ both ...
- 23.7k
-1
votes
1
answer
545
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probability mass function fitting [closed]
I have a probability mass function of some experimental data who's log looks like the following: (please ignore the fact that it is not normalized)
![alt text][1]
[image shack image removed]
(...
- 53.3k
5
votes
2
answers
368
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Reference to iterated logarithm law and Smirnov law of empirical CDF
I am reading V. Vapnik's "Statistical Learning Theory". The author layouts following two statistical laws related to empirical CDF. I am looking for reference about proofs on these two laws.
Let $...
- 253
2
votes
0
answers
366
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Convergence rate of Pearson correlation matrix
I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let $\rho(...
CommunityBot
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4
votes
3
answers
433
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Expectation of $(c+e^{N(0,\sigma^2)})^{-n},\, n>0$
I would like to know if there's a way to compute or approximate the following expectation:
$$\mathbb{E}[(c+e^X)^{-n}]$$
where $X=N(0,\sigma^2)$ and $n,c>0$ (you can also assume that $n$ is a ...
CommunityBot
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2
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1
answer
164
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Is there any parameter space of Cramér–Rao_bound
It is known that Cramér–Rao_bound is the lower bound of variance of a parameter. A useful link is https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound There is also a term called '...
CommunityBot
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0
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60
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Consistency of M-estimators when the constraint set also has to be estimated
Let $K \subset \mathbb R^n$ compact and convex. Also let $H$, $G_i, \; i \in \{1,\dotsc,m\} $: $K \to \mathbb R$ be convex functions.
Assume we have the following convex optimization problem:
$$
\...
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1
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0
answers
44
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Validating a probability density distribution forecast model for a Markov process
Let's say we have a Markov process $X_t$, and we come up with a forecast model that takes some information from outside world and says: "value $X_{t+1}$ has probability density distribution $P_t(x)$". ...
- 3,085
1
vote
2
answers
388
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Average Multivariate Gaussian
Suppose we have a (possibly infinite) collection k-variate gaussian distributions $\{(\mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda}))\}$ ($\lambda$ is just a label), and for each distribution $\mu \in [-...
- 3,085
3
votes
1
answer
724
views
Expected value (probability) maximization with binomial distribution
I need to solve an optimization problem that involves an expected value like
$$F(n,x) = \sum_{k=0}^n \binom{n}{k} p^k(1 - p)^{n - k} f(k,x).$$
Here $f(k,x)$ is actually a probability coming from a ...
- 128k
3
votes
1
answer
308
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Bounds on the probability of k-of-n events in terms of bounds on single and pairwise probabilities
Let $A_1,\dotsc,A_n$ be events in a probability space, and let $N = \sum_{i=1}^n \mathbf{1}_{A_i}$ be the random number of events that occur. For a fixed value $k \in \{1,\dotsc,n\}$, what can be ...
CommunityBot
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1
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200
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How are epidemic models simulated in case of mobility?
I am not a mathematician but out of curiosity I am trying to implement the SIS epidemic model when the nodes have mobility to understand how it will change the results. I understand how to perform ...
CommunityBot
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1
answer
258
views
Are such averages known with representations of $S_n$?
Like is there a sense in which one can quantify that for two group elements (in different conjugacy classes) their characters are "close" for some fixed irreducible representation? (feel free to stick ...
- 6,357
3
votes
2
answers
751
views
An Upper Bound for the Average of Top Order Statistics
The following problem arises when we try to bound the expected offline optimal value of a simple online assignment problem with random values and unit weights, by its deterministic approximation.
The ...
- 24.8k
0
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1
answer
408
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Generating independent random variable from two correlated random variables
Suppose two random variables $X$ and $V$ are given. I am wondering what kind of condition we need to impose on joint distribution of $V$ and $X$ to make sure that there exists a random variable $Z$ ...
CommunityBot
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12
votes
3
answers
8k
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Constructing Bernoulli random variables with prescribed correlation
For which $n \times n$ correlation matrix $C$ can one construct Bernoulli random variables $(B_1, \ldots, B_n)$ with correlation $C$ ?
Following the approach described in this MO thread, one can ...
CommunityBot
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4
votes
1
answer
2k
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variance of compound binomial distributions
The below is motivated by a problem I'm observing in my experimental data
I have m boxes, where each box is supposed to contain k molecules of mRNA. The measurement process includes labeling all the ...
- 28k
4
votes
3
answers
2k
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Correspondence between Viterbi algorithm and Smith-Waterman
Viterbi is an algorithm for finding the maximum likelihood assignment to the hidden variables of an HMM, given the observed variables (we know the transition and emission probabilities of the HMM). ...
- 303
3
votes
1
answer
306
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Mutual information decrease with coarse-graining
Let $X,A,Y,B,C,D$ be random binary variables. $D$ is independent from $X,A,C$ and $C$ is independent from $Y,B,D$.
Is it true that:
If $I(Y:B|D=0)\leq \epsilon$ then $I(X\oplus Y:A\oplus B|C=0,D=0)\...
CommunityBot
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3
votes
1
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711
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Expectation of Mahalanobis norm
Let $(g_i)_{i=1,...,d}$ sampled i.i.d. from a standard Gaussian, and $(\lambda_i)_{i=1,...,d}$ non-random s.t. $\max_i(\lambda_i)=1$ and $\lambda_i>0, \forall i$.
I am looking for the expectation ...
- 128k
2
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0
answers
175
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Implication of MGF inequality
Let X and Y be two random variables. Denote by $F_X(x)$ and $F_Y(y)$ their CDFs and by $M_X(t)$ and $M_Y(t)$ their MGFs.
It is known that X and Y have the same CDF iff they have the same MGF.
My ...
- 517
2
votes
2
answers
158
views
Do all positive distributions on $N$ variables factor pairwise?
The Hammersley-Clifford theorem says that any positive probability distribution satisfies one of the Markov properties with respect to an undirected graph G if and only if its density can be ...
- 24.8k
2
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0
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1k
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Converse for Levy's continuity theorem
Levy's continuity theorem states that, for a sequence of random variables $\{X_n\}$ with characteristic functions $\{\varphi_n(t)\}$ and a random variable $X$ with a characteristic function $\varphi(t)...
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1
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1
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207
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Computing probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s
This question came up in my research: What is the probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s?
So far I only figured out that I can do Monte ...
CommunityBot
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2
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627
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Generalized expression for balls and bins problem
$n$ number of balls are thrown randomly to $m$ number of bins, standing in a row. The balls are labeled as $1,2,3,....n$ and bins are also labeled as $1,2,3,...,m$. The probability of $i_{th}$ ball ...
- 258
12
votes
1
answer
10k
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Square root of normal distribution
Let $X$ and $Y$ be independent random variates with the same probability distribution, $P(x)$. Assuming that the product $Z=XY$ is a random variate with normal distribution, say $$f_Z(x) = \frac{1}{\...
- 128k
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0
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444
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How to decide a value of learning rate for Stochastic Gradient Descent?
I'd like to know how to decide a value of learning rate for Stochastic Gradient Descent (SGD), such as $\eta$ on the following parameter update iteration equation,
$w_{i+1} = w_i + -\eta \nabla E_n(...
1
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0
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85
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Maximum likelihood estimation with several distributions
My question concerns using Maximum likelihood to estimate unknown parameters used by several (poisson) distributions.
The parameters are the pairs $(a_1,b_1),\dots,(a_N,b_N)$, and for each pair $(i,j),...
- 11
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2
answers
1k
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PDF of the product of normal and Cauchy distributions
I am having trouble in finding out the resulting PDF of the product of normal and Cauchy distributions. It turns out that we have a general formula for calculating the PDF of product of two random ...
CommunityBot
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0
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101
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How to fit a stochastic matrix to given data.?
Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...
3
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0
answers
286
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Inequality with CDF of order statistics
here is a problem I have been struggling with for a while now. This is for a paper I am working on. Any help would be appreciated! Here we go:
Each bidder's valuation $\theta _{i},$ $i=1,...,N$, is ...
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2
answers
2k
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Applications of cohomology to probability and statistics
Are there interesting/useful applications of cohomology (and homological algebra in general) to probability and statistics, or information theory?
By "interesting/useful", I mean "not merely ...
- 22.1k
2
votes
3
answers
826
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When do binomial distributions occur?
A binomial distribution is the distribution of the number of successes of n independent, identical Bernoulli trials. What happens when the trials are dependent and the Bernoulli trials are not ...
CommunityBot
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0
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136
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What's the variance in the Six Degrees model?
Recall the six degrees of Kevin Bacon game. You can even play the game at The Oracle of Bacon, and their search works via Breadth First Search.
I interpret the punchline as saying that if I start ...
CommunityBot
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5
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1
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3k
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1-wasserstein distance v.s. total variation distance
Suppose that $\mu_1$ and $\mu_2$ are two distributions defined on $\mathbb{R}^n$ and $\gamma$ is a symmetric distribution (around $0$) on $\mathbb{R}^n$ with compact support. Let $\gamma_x$ denote the ...
- 17k
4
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1
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821
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Does bounding moments make distributions close in total variation distance?
Let $W\sim\mathcal{N}(0,\sigma^2)$ be a "reference" Gaussian random variable.
Suppose I have a set of distributions, $\mathcal{W}$, where $W_a\in\mathcal{W}$ if it satisfies the following criteria:
...
- 28k