All Questions
Tagged with pr.probability st.statistics
1,134 questions
11
votes
1
answer
435
views
(almost) statistical independence of nodes degrees in a graph
Wireless networks are typically modeled as random geometric graphs. The number of nodes $N$ in the network is drawn from a Poisson distribution with intensity $\lambda$
$$P(N = n) = \frac{\lambda^n ...
- 241
1
vote
1
answer
321
views
"Bridging" uniform and "mass" distributions
Foreword. The original formulation of this problem was inaccurate; chamomille and Didier Piau came up with a simple example which would not solve the problem in its accurate formulation. Sorry for my ...
- 5,721
1
vote
0
answers
397
views
Random Walk vs Branching process
1) Let us consider the set of all $N!$ permutations of the $N$ elements ${1, 2, . . . ,N}$. In the random state, each permutation of these elements occurs
with probability 1/N!. The probability $Pm(N)$...
- 12.8k
1
vote
1
answer
3k
views
Generating Bernoulli Correlated Random Variables with Space Decaying Correlations
Hi,
I have a set of N objects randomly distributed in a 2D physical space. Each object (i) generates a bernoulli random number (0 or 1) based on a marginal probability Pr(xi = 1) = p. These objects a ...
CommunityBot
- 1
1
vote
1
answer
1k
views
Maximums of two correlated Gaussian processes
Hi,
This question is motivated by a statistical genetics model.
Let $(x_1,y_1)$, .., $(x_N,y_N), ... $ be i.i.d. bi-variate Gaussian random variables.
The $x_i,y_i$'s are standard Gaussians, $x_i, ...
- 1,411
5
votes
0
answers
506
views
Missing mass estimate
Let $S$ be a finite set with probability distribution $P$. Define the random variable $m_i$ to be the "missing mass" after seeing $i$ iid samples from $S$ under $P$. That is, $m_i$ is the total mass ...
- 6,541
4
votes
3
answers
439
views
Probability estimates for "beans & boxes"
From a discussion with some friends, this apparently easy problem has come out; I decided to post it here, because I believe that the answer is non-trivial and the maths beneath interesting. Partial ...
CommunityBot
- 1
3
votes
1
answer
412
views
Sparse representation of a distribution with independent and correlated variables
Here's what I'm trying to do:
Imagine a probability distribution over $\mathbf{R}^2$, $P(x,y)$. I can approximate $P(x,y)$ with set of $N$ points $\{(x,y)_i\}$ drawn from $P$. By approximate, I mean ...
- 1,902
10
votes
0
answers
391
views
Question from an economist: solving a model of traders' behavior with expectations about the future values of the variable they are currently optimizing
Motivation
I am an economist writing a paper for an academic finance journal. My paper is about the behavior of currency traders, who choose the price at which they will sell currency today, based on ...
- 101
0
votes
1
answer
915
views
Can you interpret this divergent integral?
In this ArXiv paper by Wilk and Wlodarczyk (published in Physical Review Letters), equation 16 has essentially the following definition of a function:
$$\text{f(x)=}\frac{c}{2Dx^2}\exp[\int^x_0 \frac{\...
- 5,721
1
vote
2
answers
791
views
Likelihood function for sequential random variables
Context
Consider the following sequential data generating process for $Y_1$, $Y_2$, $Y_3$. (By sequential I mean that we generate $Y_1$, $Y_2$, $Y_3$ in sequence.):
$Y_1 = X_1^' \beta + \epsilon_1$
...
CommunityBot
- 1
2
votes
1
answer
2k
views
deriving angular central gaussian distribution from a multivariate normal distribution
The angular central Gaussian (ACG) distribution on $(p-1)$-dimensional sphere $\mathbb{S}^{p-1}$ for a symmetric positive definite parameter matrix $\mathbf{A}$ is defined as
$$f(\mathbf{x},\mathbf{A}...
- 231
2
votes
2
answers
955
views
Probability calculation, system uptime, likelihood of occurence.
A little stumped! This is probably a very basic probability question, but I am lost.
At work I was asked the probability of a user hitting an outage on the website. I have some following metrics. ...
CommunityBot
- 1
18
votes
2
answers
4k
views
When is the function of a median closer to the median of the function than the mean of the function is to the function of the mean?
Background
notation: RV= random variable, $\mu=$ mean $m=$ median
Jensen's Inequality considers the relationship between the mean of a function of an RV and the function of the mean of an RV.
If $f(...
CommunityBot
- 1
1
vote
0
answers
466
views
Bounding point-wise maximum of the absolute difference of two convex functions
Let $\Delta: R \times R \rightarrow R_{+}$ be a positive and convex function (convex in, say, both the arguments) called the loss function.
Let $x \in R^d$. Moreover, let $H_1,...,H_r$ be sets of ...
- 11
1
vote
2
answers
1k
views
what will be the distribution of ratio of correlated gamma distributed random variables?
If $X\sim \Gamma(a,\sigma_x^2)$ and $Y\sim \Gamma(b,\sigma_y^2)$. What will be the probability density function of R? Where $R=\frac{X+C}{X+Y}$, here $C$ is a positive constant, $\Gamma(.,.)$ denotes ...
- 5,721
1
vote
1
answer
434
views
A point process for modeling location of trees in an infinite forest?
I am looking for an example of a stationary, infinite point process on $\mathbb R^n$ with a few simple properties. I would not be surprised to discover that there is a well-studied, canonical process ...
- 1,072
5
votes
2
answers
2k
views
Process for a Gamma distribution with non integer shape parameter
I am sampling the distribution of lifetimes of computers participating in massive volunteer computing initiatives (BOINC projects). While a phenomenological Weibull distribution makes a good ...
- 1,468
7
votes
1
answer
5k
views
Parametric vs Non-parametric Estimation of Quantiles
Motivation
Suppose that we need to estimate the median from a normal distribution with known variance. One non-parametric approach is to use the sample median as an estimator. However, this does not ...
CommunityBot
- 1
1
vote
2
answers
744
views
Order statistics: probability random variable is k-th out of n when ordered.
Given a random variable $X_1$ drawn from a distribution with cdf $F$, and random variables $X_2, \cdots,X_n$ drawn from another distribution with cdf $G$, what is the formula for the probability that $...
- 1,468
7
votes
1
answer
804
views
Random, Linear, Homogeneous Difference Equations and Time Integration Methods for ODEs
Most methods (that I know of) of numerically approximating the solution of ODEs are "general linear methods". For this type of method, the so-called 'linear stability' is examined by applying the ...
CommunityBot
- 1
3
votes
1
answer
663
views
Stationary non-isotropic spatial stochastic processes
I asked this question in math.stackexchange but got no response;
Are there any interesting examples of second order stationary processes on ${\mathcal R}^2$ or ${\mathcal R}^3$ that are not isotropic?...
- 549
3
votes
2
answers
2k
views
Why is Beta the maximum entropy distribution over Bernoulli's parameter?
Why is Beta(1,1) the maximum entropy distribution over the bias of a coin expressed as a probability given that:
If we express the bias as odds (which is over the support $[0, \infty)$), then Beta-...
- 1
0
votes
1
answer
801
views
Information criteria for ridge regression
Hi -- is there any analogue or adjustment of, say, Schwartz Bayesian (or other) information criterion that would be applicable to model selection in ridge regression with a given ridge parameter $\eta$...
- 2,791
1
vote
1
answer
294
views
Stability of discrete queue (new twist)
Hi, I am new to queueing theory. I am interested in a question that I feel should be fairly basic, yet I haven’t really found a clear solution to it. Hopefully somebody here can help me.
We have a ...
- 501
0
votes
0
answers
319
views
Estimating a multinomial sum
I have the following sum
\begin{equation}
\sum_{r_1=q+1}^{\tau}\dots\sum_{r_\lambda=q+1}^{\tau}{\tau\choose r_1,\dots,r_\lambda,\tau-r_1-\dots -r_\lambda} (\Lambda-\lambda)^{\tau-r_1-\dots-r_\lambda}
\...
1
vote
1
answer
356
views
Statistical inequality
Let $X$ be a finite discrete variable and $X\ge0$. Is it true that
$$16\operatorname{Var}(X) \le \left[8{\mathbb E}(X) + \operatorname{Range}(X)\right]\operatorname{Range}(X)$$
where $\operatorname{...
- 303
0
votes
2
answers
339
views
Efficient Method for Calculating the Probability of a Set of Outcomes?
Let's say I'm playing N different independent "games". For each game, I know the probability of winning, the probability of tying, and the probability of losing.
From these values, I've also ...
- 29.8k
2
votes
1
answer
250
views
Expectation of RVs with Poisson-type decay
I need to bound the expectation of a nonnegative random variable that satisfies a Poisson-type tail bound:
$\mathbb{P}( X \geq t ) \leq \min( d \cdot (\frac{a}{t} )^{t}, \ 1)$ for $t > 0$
where $...
- 23
1
vote
2
answers
175
views
is there an interpretation to the inverse of $I-M$ in multitype branching process, where $M$ is the mean matrix?
Assume we have a multitype branching process, i.e., we have a mean matrix $M_{ij}$ and $M_{ij}$ is the expected count of generating $j$ from $i$ in one time step, i.e.:
$M_{ij} = \sum_{r} n(r,j)P(r | ...
- 710
3
votes
2
answers
453
views
What is this probability distribution?
Suppose we have a family $F_0,F_1,\dots$ of independent random variables which take the value $1$ with probability $p$ and $0$ otherwise; let $\delta$ be a number between $0$ and $1$. Let
$X_n = \...
- 1,180
3
votes
2
answers
1k
views
Tightness of probabilty distributions
Let $\mathcal{P}(\mathbb{N})$ be the set of all probability mass functions on $\mathbb{N}=\{1,2,\dots \}$. Let $E$ be a closed(with respect to pointwise convergence, or equivalently the total ...
- 413
4
votes
2
answers
1k
views
expected values over binomial distributions
In some works of economics/risk analysis etc., I have seen situations where people take the expected value of a function (such as a utility function/cost function) over a binomial distribution:
$$F(n)...
- 3,283
32
votes
5
answers
2k
views
You pass X people and Y people pass you: how relatively fast are you?
This question occurs to me every time I go jogging. I suspect every runner probabilist in the world must have thought of it (though I'm no probabilist), but I could not specifically find it online. I ...
- 1,391
1
vote
1
answer
242
views
Measuring the randomness in random numbers
I'm looking to write a program to investigate a few random number algorithms. Basically I am looking to see if the spread of numbers is indeed randomly distributed enough. What kind of statistical ...
- 173
5
votes
2
answers
1k
views
Inequality involving probability measures [closed]
I have been working on a problem(alternate minimization) where I want to establish an inequality in which I am stuck.
An $\alpha$- parameterized version of the divergence(Kullback-Leibler) takes the ...
- 779
21
votes
1
answer
3k
views
Intuitive Proof of Cramer's Decomposition Theorem
Cramer's decomposition theorem states that if $X$ and $Y$ are independent real random variables and $X+Y$ has normal distribution, then both $X$ and $Y$ are normally distributed. I've seen a few ...
- 2,071
3
votes
1
answer
578
views
Why doesn't Stein effect happen for multinomial distributions?
(Medeen, et all, 1998)" show that Maximum Likelihood estimate is admissible for multinomial distribution under squared error. On other hand, James and Stein showed that arithmetic average is not an ...
- 2,791
2
votes
1
answer
419
views
Approximation of the law of a stochastic process
Hello Dear fellows,
I thank you in advance for your help and ideas.
I have just read an article and want you to help me understand the rational behind a part of it.
We have two processes $v_t$ and $...
CommunityBot
- 1
4
votes
2
answers
2k
views
Elo Rating System Help with the Maths around number of matches
I'm creating a system that will allow people to rate images.
My idea is to use an Elo Rating system (http://en.wikipedia.org/wiki/Elo_rating_system) for each image and then use crowdsourcing to have ...
- 2,401
3
votes
2
answers
921
views
Characteristic operator
Let $X_t\in\mathbb{R}$ be an Ito diffusion process given by $$ dX_t=a(b-X_t)dt+\sigma dW_t$$, then the characteristic operator of $X_t$ is given by $$L=a(b-x)\frac{\partial}{\partial x}+\frac{\sigma^...
- 498
0
votes
0
answers
138
views
Why do I not use post hoc tests with a 2 x 2 factorial?
I know this is an obvious answer. I am probably over thinking what I'm doing, but I cannot recall. Does it have to do with not having enough variables to compare the various means?
2
votes
0
answers
530
views
About generalization of stirling numbers of the second kind
Hello,
The Stirling numbers of the second kind count how many ways can a set of $k$ elements be partitioned into $n$ non-empty classes, with $k=n,n+1,\dots$.
My question is: Is there a ...
0
votes
1
answer
284
views
The density of x_1^n+x_2^n where x_i are Gaussian
We define a process $\chi_k^n=\sum _{i=1}^k x_i^n$ where x_i are iid gaussian processes.
I try to find the distribution of $\chi_k^n$. If k=1 then we get $f(x^n=y)=\frac1n y^{\frac{1-n}{n}}\exp(-y^{2/...
CommunityBot
- 1
5
votes
5
answers
3k
views
Computing correlation between time series with missing data.
Suppose you have two simple Ar[1] series of the form $y_n=y_{n-1}+e_n$ and $x_n=x_{n-1}+m_n$, where $e_n$ and $m_n$ are normal white noise processes with no auto-correlation and $Corr(e_n,m_n)=p$. ...
- 151k
1
vote
3
answers
291
views
Is any bias introduced from initial clustering
I hope this is an appropriate forum for this question, and I asked on math.stackexchange as well. If it doesn't belong, I don't mind closing this. If my questions is not clear, please just let me ...
- 168
1
vote
3
answers
332
views
Is ERNIE output skewed by statistical tests?
ERNIE is a hardware random number generator used to generate winning Premium Bond numbers in the UK. Wikipedia says: "ERNIE's output is independently tested each month by an independent actuary ...
- 2,401
0
votes
1
answer
1k
views
Kernel width in Kernel density estimation
Hi,
I am doing some Kernel density estimation, with a weighted points set (ie., each sample has a weight which is not necessary one), in N dimensions.
Also, these samples are just in a metric space (...
- 51
2
votes
0
answers
548
views
What will be the distribution of harmonic mean of two correlated gamma random variables?
Suppose there are two correlated random variables $X_1$ and $X_2$ both are gamma distributed but having different shape and scale parameters with correlation coefficient $\rho$. What will be the ...
- 133
3
votes
1
answer
2k
views
sum of order statistics
Suppose I have N real random variables with identical PDF. At every instance of these r.vs, I pick $K$ largest out of $N$. Lets call their sum as $S_K$. Alternatively, based on some criteria, I ...
- 3,937