A probability provides a quantitative description of the likely occurrence of a particular event.
8
votes
0answers
108 views
Product of two independent random variables
I have a sample of about 1000 values. These data are obtained from the product of two independent random variables $\xi \ast \psi $. The first random variable has a uniform distribution $\xi \sim ...
5
votes
0answers
118 views
Luce Choice Axiom, question about conditional probability
I'm reading Luce (1959). Then I found this statement:
When a person chooses among alternatives, very often their responses
appear to be governed by probabilities that are conditioned on the
...
5
votes
0answers
113 views
How to calculate the probability of absence for a certain category of artefacts from a sample, given prior knowledge about its abundance?
In archaeology, artefacts are commonly classified in categories according to certain criteria (those may include manufacturing technique, decoration, function, chronology, etc).
I am trying to ...
5
votes
0answers
219 views
Marginal distribution of the diagonal of an inverse Wishart distributed matrix
Suppose $X\sim InvWishart(\nu, \Sigma_0)$. I'm interested in the marginal distribution of the diagonal elements $diag(X) = (x_{11}, \dots, x_{pp})$. There are a few simple results on the distribution ...
4
votes
0answers
36 views
Estimating repeat shoppers from an incomplete sampling
I'm trying to estimate how many people visited the farmers market once, twice, thrice, etc. in a given time period, using sampled data. We have interview data from approximately 50% of visitors as ...
4
votes
0answers
120 views
Conditional probability update for correlated Poisson variables
Some background:
I am trying to estimate the number of failures in two related machine populations. I model machine failures in a year as a correlated Poisson process as such:
$Y_0,\ Y_1$ and $Y_2$ ...
4
votes
0answers
50 views
How to pick from a set of random variables the one with the highest mean using a fixed number measurements?
Suppose we have $N$ approximately normally distributed continuous random variables $X_1, X_2, X_3, \ldots,X_N$, each with an unknown mean and variance. I'd like to find the random variable with the ...
4
votes
0answers
65 views
Covariance of INID order statistics
In the IID case, it is known that all order statistics are positively correlated.* Thus, we know that $$\text{Cov}(X_{(i)},X_{(j)}) \geq 0.$$ Is this known in the INID (independent, non-identically ...
4
votes
0answers
120 views
How to predict the probability of a performance rank for a test taker on a future test based on previous test scores?
Four students sit a series of tests, each test contains 10 questions. Students are ranked 1-4 in each test depending upon who got the most questions correct.
...
4
votes
0answers
127 views
Predicting time interval based on statistics
Let's say, we have some random event. We also have a hist of time intervals between two events, based on statistical data. For example, a frequency distribution:
...
4
votes
0answers
734 views
What is the difference between deterministic models and probabilistic models that assume that error follows a degenerate distribution?
I have thought for a while about this, and I'm not entirely certain what the difference is between a deterministic model and a probabilistic model with error following a degenerate distribution ...
3
votes
0answers
58 views
Distribution of variable
How to find the distribution of $$\sum_{i=1}^n (X_i - X_{1:n}),$$ where $X_i$ are i.i.d. random variables and $X_{1:n} = \min(X_1,X_2,...,X_n)$?
I need to find the distribution in a particular case, ...
3
votes
0answers
85 views
Is my Theory on the Law of Truly Large Numbers and Ancestors Valid?
I am an anthropology student. I am researching the inter-relatedness of most if not all humans as part of my studies. I personally have an interest in statistics and probability theory and I have used ...
3
votes
0answers
296 views
Why does a finite, irreducible and aperiodic Markov chain with a doubly-stochastic matrix P have a uniform limiting distribution?
The theorem is "If a transition matrix for an irreducible Markov chain with a finite state space S is doubly
stochastic, its (unique) invariant measure is uniform over S."
If a Markov Chain has a ...
3
votes
0answers
73 views
Estimating how quickly people submit their answers
In a contest for solving a problem, a person submits his/her answer for the problem at any time during the contest. The contest system provides the person with feedback on the submitted answer. The ...
3
votes
0answers
147 views
Probability distribution for transformation of a random variable
Let $g(x)=1$ if $x \leq c$ and $g(x)=(1-x)/(1-c)$, where $0 \leq x \leq 1$ and $0 <c <1$.
So $g$ is an non-increasing function.
Define $g^{-1}(y)=\inf\{0 \leq x \leq 1 \mid g(x) \leq y\}$.
...
3
votes
0answers
133 views
$\sigma$-algebra intersection of infinite subsets
I am working out a book on Lebesgue measure by Bartle, and would like to see the steps that go into the construction of a proof for the following:
Show that any $\sigma$-algebra of subsets of ...
3
votes
0answers
45 views
Comparing rates of events between an exposed and unexposed group
I'm looking at 2 papers which are addressing the same issue using a retrospective cohort study design but use different methodology to analyse data and I would like to know why the methods used ...
3
votes
0answers
164 views
Normalization/comparison-when the observed cardinality is a random variable, over finite sets
Given, $p_{ij}=\frac{|A_i|+|A_j|}{|A_i\cup A_j|}$ for sets $A_i$,$A_j$ $\forall i\not=j ,\forall i \in \left \{ 1,2...n \right \} $ and given the fact that $|A_i \cap A_j|>0$ for all $A_i,A_j$: ...
3
votes
0answers
142 views
Methodology for validation of stochastic simulations with Kolmogorov-Smirnov test
I'm a phd student in Geography, i need some help (or good ressources) to understand why and when i need to use PIT (Probability integral transform) in my validation program for simulation.
I explain ...
3
votes
0answers
100 views
Calculating probabilities related to order statistics
Crux of the question
Let $q \sim F$ with support $[0,1]$. Let $q_j$ be the $j$th order statistic of $N$ draws from $F$. Let $z_j \sim \text{Bernoulli}(q_j)$. See that these draws are independent, ...
2
votes
0answers
24 views
+100
Importance sampling of finite path of stochastic difference equation
Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...
2
votes
0answers
43 views
A question regarding the central limit theorem
I would appreciate if you could please take a look at the following attempt. I am preparing for an upcoming exam, and this is taken from a past exam paper. I would like to know if it is correct. Thank ...
2
votes
0answers
46 views
Using probability scores from a random forest
I've the following problem. I've a data set that tries to predict whether a given buy event will happen or not (0/1) when a customer sees a certain product, and I've features created for both the ...
2
votes
0answers
49 views
Estimator bias without a closed form?
Given a regression loss function $l(Z,\beta)=||Y-Z\beta||_2 + \lambda \beta^TD\beta + r(X,Z)$ where $X$ is the predictor matrix, I would like to estimate a $Z$ that minimizes the above loss in a ...
2
votes
0answers
61 views
Probability output from support vector machine (svm) with soft margin
Based on my very simple understanding of SVMs, it seems like a probabilistic output would be a very useful thing to have. Soft margin seems to part of the way toward accounting for noisy data, but ...
2
votes
0answers
78 views
How do I calculate the number of combinations, where a minimum number of elements must be different?
I have 54 names.
I would like to know how many unique combinations of 6 names I can create, where at least 2 of the names are different from those in every other combination?
My maths is extremely ...
2
votes
0answers
40 views
Order statistics of equal correlated continuous random variables
Suppose that $X_1, \ldots, X_n$ are mutlivariate normal with equal correlation
$\rho$ and each of them are marginally
distributed as $N(0,1)$. Let $X_{(1)}, \ldots, X_{(n)}$
be the corresponding order ...
2
votes
0answers
35 views
In a random selection of addresses, determine if users are unable to access one of the addresses
Our client software randomly selects a server to connect to. There are four separate addresses the software randomly chooses from. Two addresses are "old" and two are "new". As many firms need to ...
2
votes
0answers
25 views
Estimating the likelihood of independence of two discrete variables using the co-occurrence count matrix
I have some data about users from different regions visiting different directories of some website. Aggregating that data I get the co-occurrence frequency matrix (for regions and directories). Now I ...
2
votes
0answers
104 views
Kullback-Leibler vs Hellinger Distance
I am working on this problem in which I have a dataset of n-dimensional examples that come from different and unknown distributions. Given a new sample, I wish to find k examples from the dataset that ...
2
votes
0answers
67 views
Distributions similar to the family of stable distributions
Are there any other distributions with similar properties to the family of stable distributions? That is, $\alpha$-stable, normal tempered stable, classical tempered stable, etc. etc. where the ...
2
votes
0answers
89 views
Collapsibility: Odds Ratios versus Risk Ratios
It is known that odds ratios enjoy a certain symmetry. For example, the odds ratio of outcome $Y$ is the inverse of the odds ratio of outcome $\neg Y$. Risk ratios, on the other hand, do not enjoy ...
2
votes
0answers
56 views
Distribution/expected length of the shortest path in infinite random geometric graphs
Consider an infinite random geometric graph $G(\rho,d)$ in which vertices are uniformly and independently scattered over the 2D plane with density $\rho$ and edges connect the vertices that are closer ...
2
votes
0answers
67 views
Random sampling for estimating mutual information - Time complexity and sampling error?
I have a dataset and I want to compute the mutual information (MI) for a selected set of variables. The dataset is large enough so that computation of the MI may take undesirably long time. Can I just ...
2
votes
0answers
45 views
Dealing with dependant data when estimating probability of an event happening
I have 10 year worth of data from 1970 to 1980 (40 quarters).
For each quarter I have five measurements M1, M2, M3, M4 and M5.
TWIST: Although the data I have is on individual patient level the ...
2
votes
0answers
177 views
How to find principal components from variance-covariance matrix?
If I have the following var-cov matrix:
$\Sigma_{A,B,C} = \left(\begin{array}{ccc}
1 & 2 & 3 \\
2 & 4 & 6 \\
3 & 6 & 9 \\
\end{array}\right)$
(1) How can I find a constant a, ...
2
votes
0answers
130 views
Calibration for random forests
I want to evaluate the calibration of the random forest using val.prob (rms package, R). I have no problems using it and getting an output, but I feel the results may not be accurate because I don't ...
2
votes
0answers
117 views
Name for $E[X]^2/E[X^2]$?
Has anyone seen the following quantity come up in the literature?
$$\frac{\mathbb{E}[X]^2}{\mathbb{E}[X^2]}$$
I saw it in equation 10 of this paper.
2
votes
0answers
55 views
Conditional distribution of quadratic forms
Given that $Y$ follows multivariate normal distribution ,i.e, $N_n (0, \sigma^2 I_n)$, we want to find the distribution of $Y'Y$ given that $a'Y=0$ where $a$ is a non zero constant vector.
I know ...
2
votes
0answers
47 views
Multi-information of a uniformly distributed random variable on the L1 sphere
I posted this question in the stackexchange mathematics forum without any reponse. Maybe it was the wrong forum, so I try it here.
I tried to compute the multi-information (MI) $I[\mathbf U] = ...
2
votes
0answers
33 views
Sampling from elliptic pde solution in high dimensions
What is known about sampling from solutions to elliptic PDE's in high dimensions, where it is computationally infeasible to construct or store the actual solution?
For example, let $u$ solve the ...
2
votes
0answers
56 views
Metric for ordinal, non-normal distribution in Markov models
I am working with Markov models with ordinal, non-normal distributions of probabilities. Ultimately, I would like to create a metric for determining the probability of a specific path occurring. ...
2
votes
0answers
36 views
How to improve estimation of a deconvolved density
I have the following problem:
Y = X + e
with
Y = Total reaction time (noisy signal)
X = selection time (signal)
e = discrimination time (noise)
I am interestend in the distribution for X and ...
2
votes
0answers
14 views
Maximizing choice
There are N number of people and X amount of objects with different values. Each person will choose an object and will obtain that objects value. If multiple people choose the same object then the ...
2
votes
0answers
38 views
Exchangeable Processes over the Simplex
You are likely all familiar with Polya Urn process. I initially start with an urn containing $b$ black balls and $w$ white balls. At each step, I sample a black ball with probability $\frac{b}{b+w}$ ...
2
votes
0answers
26 views
The product distribution: how fast does dissimilarity increase as a function of number of samples?
If $\mathcal{D}$ is a distribution, let $\mathcal{D}^n$ denote the $n$-fold Cartesian product of $\mathcal{D}$. In other words, $\mathcal{D}^n$ is the distribution of $n$-tuples $(x_1,\dots,x_n)$ ...
2
votes
0answers
103 views
Predicting percent of male and female party attendees
Problem #1: easy, probably solved
Suppose someone holds a party (A) and invites $n_A$ people. You're not attending a party, but you get full list of guests' identity card and are told that out of ...
2
votes
0answers
64 views
Modeling a 1D random walk with nonconstant probability in a point
There is a 1D discrete random walk system which the probability of all points are 1/2(probability of going forward and backward) except one point which locate on l (l is an integer number).the ...
2
votes
0answers
88 views
Joint distribution of two sums of correlated variables
Suppose that $(X_1, Y_1)$ and $(X_2, Y_2)$ are independent and have the same joint distribution $F_{X,Y}$, which is a known copula $C_{X,Y}(F(X), F(Y))$. Also, suppose that $V = X_1 + X_2$ and $W = ...
