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Questions tagged [probability]

A probability provides a quantitative description of the likely occurrence of a particular event.

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2k views

Probability inequalities

I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts. My problem is to find an exponential upper ...
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866 views

Rademacher complexity of logistic regression

Consider logistic regression. We have the logistic loss function, $\phi: R\rightarrow [0,1], \phi(u)=\log(1+\exp(-u))$, which is Lipschitz, and we have the linear function class $F=\{f_w:R^d \...
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173 views

Intuitive understanding of the Halmos-Savage theorem

The Halmos-Savage theorem says that for a dominated statistical model $(\Omega, \mathscr A, \mathscr P)$ a statistic $T: (\Omega, \mathscr A, \mathscr P)\to(\Omega', \mathscr A')$ is sufficient if (...
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156 views

Is probability fundamentally about reference classes (real or imagined)?

Question: It seems that frequentism and Bayesianism may not really be different as far as the the ultimate basis for what a probability is (relative frequency within a reference class) - it's just ...
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312 views

Sum of absolute values of T random variables

Where X is a r.v. following a symmetric T distribution with 0 mean and tail parameter $\alpha$. I am looking for the distribution of the n-summed variable $ \sum_{1 \leq i \leq n}|x_i|$. $Y=|X|$ ...
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412 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 ...
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115 views

Combined distribution of beta and uniform variables

Given $$X \sim \text{Beta}(\alpha,\beta)$$ (where $\alpha=\beta$, if that helps) and $$\theta \sim \text{Uniform}(0, \pi/2).$$ I'm trying to find a formula for $P(Y)$ (or even the cdf) of $$Y = X +...
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96 views

Is there a name for, and some significance of, $\mathbb E e^{X^T \Omega X}$?

Let $X \in \mathbb R^r$ be a random vector. The moment generating function (MGF) $m(t) = \mathbb E e^{X^Tt}$ is ubiquitous in statistics and probability. In my research, I've had to deal with the ...
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144 views

Time evolution of a Bayesian posterior

I have a question regarding the time evolution of a quantity related to a Bayesian posterior. Suppose we have binary parameter space $\{ s_1, s_2 \}$ with prior $(p, 1-p)$, The data generating ...
6
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156 views

Confidence interval of the third moment of normal distribution

How to compute exact confidence interval for the third moment of normal distribution $N(a, \sigma^2)$?
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169 views

Multinomial distribution conditional on number of distinct items

I want to sample from the integers $\{1, \dots, k\}$ with probabilities $\{ p_i \}_{i=1}^k$, with replacement, until I see $m$ distinct elements (call that $n$ times). You can view the distribution I ...
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950 views

Reorder point with stochastic lead time and demand

I'm trying to determine the optimal reorder point for some products. The reorder point must be greater than the demand during lead time a $\%$ of the times that I should determine, let's say $95\%$. ...
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2k 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 ...
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333 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 $Y_2$ ...
5
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338 views

Conditional expectation in the multivariate normal distribution

Suppose $(X_1, X_2, X_3)^T$ is multivariate normal. What is the conditional expectation $E(X_1 \mid X_2, I(X_3 > 0))$? Here, $I(X_3>0)$ is the random variable that takes the value one when $...
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51 views

Unbiased estimator for top-k bernoullis

Supposed I have $n$ coins and I'm interested in finding the $k < n$ coins which have the highest odds of coming up heads and I want to know $p(heads)$ for each of these $k$ coins. Assume that I'm ...
5
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0answers
190 views

Probability with an unbalanced coin where consecutive flips are not independent

Thanks in advance for the help. Suppose someone has an unbalanced coin that they flip 100,000 or so times in a row. This person then gives you the results. You do not know the probability of ...
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431 views

Expectation of a strictly increasing function

Assume that $X_1$ and $X_2$ are two i.i.d. random variables with pdf $f$. Also, assume that $a$ and $b$ are two fixed real numbers such that $a>b$. If $g$ is a strictly increasing function, do I ...
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193 views

Suggestions for a recent book on probability

I've been dealing with statistics for a few years now. Up to now, for the probability part I've been referring to my old university book (my edition is even older, by the way), and of course the ...
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85 views

How can I calculate the probability that the product of two independent random variables does not exceed $L$?

I have one variable, $X$, which is provided hourly for a period of one month (720 total values in the series). I have another variable, $Y$, which is provided quarterly (for which I am provided the ...
5
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49 views

Why is the standard deviation the error on the singular measurement?

I'm a beginner with the study in data analysis in Physics. I'm trying to understand the meaning, in the field of experimental Physics, of the standard deviation $\sigma$ of a series of data. There ...
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214 views

I'm not asking for a conjugate prior. Is there a distribution $p(x|y)$ that satisfies $\int p(x|y)Beta(y|a,b) dy = Beta(x| a', b')$?

I know the result of integrating a Gaussian against another Gaussian is still Gaussian, $$\int N(x|\mu_y,\sigma_y)N(y|\mu,\sigma) dy = N(x|\mu',\sigma')\quad.$$ Can I get the same form for Beta ...
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1k views

Combining multiple classifiers

I am trying to do a binary classification of text articles into {relevant, non-relevant}. The text articles have following features: [[article text, ...
5
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0answers
391 views

Marginal probability function of the Dirichlet-Multinomial distribution

I can't seem to find a written out derivation for the marginal probability function of the compound Dirichlet-Multinomial distribution, though the mean and variance/covariance of the margins seem to ...
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0answers
103 views

Uniform Convergence of Moment under Empirical distribution

Let $X$ be standard Gaussian random variable with cdf $F(x)$. Let $\{X_i\}_{i=1}^n$ be a sequence of i.i.d. standard Gaussian random variables. And let $F_n(x)=\frac{1}{n}\sum_{i=1}^n1_{\{X_i\leq x\}}$...
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510 views

logistic regression prediction: changing interpretation with changing prior

The data include 3 equally sized subsets A, B and C, belonging to two classes: A belongs to class 1. B and C belong to class 2. The prior probabilities of an observation coming from class 1 ...
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977 views

The distribution of STD/MAD for a Student-t

Where $X \sim a$ symmetric Student-t Distribution $t_\alpha$, with power law tail $\alpha>2$, looking for the distribution of $$ \frac{\sqrt{ \sum_{i=1}^n x_i^2 }}{\sum_{i=1}^n |x_i|}, $$ in ...
5
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0answers
227 views

Similarity algorithms

Let's say I have 300 restaurants that I want to compare to each other on the basis of a "similarity score". To try and determine similarity scores, I pick a reference restaurant and pick 3 other ...
5
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0answers
175 views

Combining evidence using Dempster-Shafer theory

Can someone post a simple explanation of Dempster-Shafer theory? There are lot of links available but the reading material in those sites is academic in nature and time consuming to read and ...
5
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0answers
135 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 ...
5
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0answers
124 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 $\frac{b}{b+w}$ ...
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598 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: ...
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0answers
58 views

Proving an inequality for CDF's

I am working on a proof to show that given $x_1, x_2,\ldots,x_k$ random variables with a joint pdf and joint CDF, show that $$ 1-\sum_{i=1}^k \overline{F_i(x_i)} \leq F(x_1,x_2,\ldots,x_k) \leq \min_i ...
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0answers
51 views

Survival probability of a random walk with renewal timings

A random walker starting at time $t=0$ and location $x=0$ moves to the right ($x+1$) or the left ($x-1$). The $k^{\mathrm{th}}$ moves to the right and left occure at the times $\sum_{i=1}^{k} R_i$ and ...
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0answers
38 views

What is the probability of forming a cluster of the size M given N random points?

Suppose we generate a sample of uniformly distributed numbers of the size $N$ in the range $[0, 1]$: $x \sim U[0, 1]$. We consider that if the difference between any 2 numbers is smaller than some ...
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55 views

Predicting a Twitter user from individual tweets probabilities

Let's say I have three tweets and those three tweets are all from either Mary or John. There is no possibility for mixed result. ...
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0answers
49 views

Independence of random vectors

Let $\mathbf{X}=(X_1,X_2,\cdots, X_n)$ and $\mathbf{Y}=(Y_1,Y_2,\cdots, Y_m)$ be two random vectors. If each component of $\mathbf{X}$ is independent of $\mathbf{Y}$ can we say that $\mathbf{X}$ and $\...
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0answers
38 views

A generalized boy or girl problem

Consider a sample $N$ numbers, independently drawn from a discrete uniform distribution over $\{1,2,...,k\}$, with replacement. Suppose I know that at least one of the $N$ numbers is 7 (like the sex ...
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0answers
65 views

Is it a problem that limiting frequencies (can) violate countable additivity?

I`ve stumbled upon the following paper by Alán Hajek https://www.jstor.org/stable/40267419, in which the author states that the Frequentist interpretation of probabilities as limiting frequencies ...
4
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0answers
91 views

Uniform convergence of expectation?

Let $U_i$, $i = 1 ,2\dots, $ be i.i.d. standard normal random variables. For every $n$, let $f_n:\Theta \times \mathbb R^{n} \to [0, 1]$, where $\Theta$ is some compact subset of $\mathbb R$ [may ...
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0answers
34 views

Can someone clearly paraphrase the following argument?

Metaculus is a site where users make and justify predictions on various questions. My question is about an estimation of the probability that a human will live to 120 years by the year 2024. I am ...
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0answers
170 views

Does $\cos(U)$ have the same distribution as $\sin(U)$, when $U \in (0, 2\pi)$?

Consider an uniformly distributed variable $U$ in $(0,2\pi)$. My impression is that $\cos(U)$ have the same distribution as $\sin(U)$. Is my assumption correct?
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0answers
57 views

An arithmetic mean preserves normal distributions, maximum preserves Frechet/Gumbel/Extreme Value distributions, but what about all other power means?

Let the $k$-power mean of two numbers $x$ and $y$ be defined as $M^k(x,y) = \left(\frac{x^k+y^k}{2}\right)^{1/k}$. For the case $k=1$, we have that if $X,Y$ are independently normally distributed, ...
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166 views

Are there any (commonly used/seriously proposed) discrete distributions that are “pathological”?

An example of a commonly used "pathological" continuous distribution is the Cauchy distribution, which is considered to be "pathological" as it has no defined mean or variance. It can be defined as ...
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0answers
105 views

Does this sampling without replacement have a name?

There can be many ways and ad hoc variants to perform sampling without replacement from a limited population. Consider we have $k$ categories (types of objects) and the k-length vector of their ...
4
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0answers
435 views

Standard normal quantile approximation

In the book Asymptotic theory of statistics and probability by Anirban DasGupta (2008, Springer Science & Business Media) in page 109 Example 8.13 I found the following approximation $$\Phi^{-1}\...
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0answers
73 views

Extension of probability on different algebras

I have a probability measure $P$ defined on 2 semi-algebras $S_1$ and $S_2$. According to the first extension theorem, there is a probability measure $P^*$ defined on the smallest algebra that ...
4
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0answers
84 views

Trading signals example from Marginal Revolution blog

From http://marginalrevolution.com/marginalrevolution/2016/09/someone-give-doug-blog.html: Many trading signals reliably predict prices, but not strongly enough to overcome transaction costs (i.e....
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0answers
50 views

Are there statistical tests for detecting model miscalibration that can be used as an alternative to binning probabilities?

I have a model which gives success probability estimates for a sequence of non-identically distributed Bernoulli trials. When the model is not constrained to be well calibrated with respect to ...
4
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0answers
461 views

Expectation and variance of sample mean with random sample size

I have a question regarding sampling where the sample size itself is a random variable. Say I have two sub-populations $A$ and $B$ from which I can sample a real valued random variable with ...