Questions tagged [probability]

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

2,145 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
36
votes
2answers
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 ...
25
votes
1answer
534 views

Jaynes' $A_p$ distribution

In Jaynes' book "Probability Theory: The Logic of Science", Jaynes has a chapter (Ch 18) entitled "The $A_p$ distribution and rule of succession" in which he introduces the idea of $A_p$ distributions,...
11
votes
0answers
1k 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 \...
8
votes
0answers
173 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 ...
7
votes
0answers
387 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|$. $Y=|X|$ ...
7
votes
1answer
384 views

Markov Switching Forecast. How can I derive this?

Consider the autoregressive model, $\left[ \begin{array}{l} y^{\ast}_t\\ x_t^{\ast} \end{array} \right] = \left[ \begin{array}{l} a_{11}\\ a_{21} \end{array} \begin{array}{l} a_{12}\\ a_{...
7
votes
0answers
422 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 ...
7
votes
0answers
350 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 $Y_2$ ...
6
votes
0answers
140 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 +...
6
votes
1answer
128 views

Help understanding a paragraph in Kadane's book Principles of uncertainty

Consider two infinite sequences of indicators of events, $s_1$ and $s_2$, with respective relative frequencies $l_1$ and $l_2$, where $l_1\neq l_2$. Let $A$ be the indicator of an event not an ...
6
votes
0answers
397 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 $...
6
votes
0answers
100 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 ...
6
votes
0answers
153 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
votes
0answers
576 views

Exercise on Borel Cantelli Lemma ($\limsup X_n/ \ln(n) =1$ a.s.) help required to rigorously write the statement

I hope this question is within the scope of this site. Please note that I have solved this Exercise, I do have doubts about my presentation though and about how to rigorously empathize on the ...
6
votes
0answers
184 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 ...
6
votes
0answers
493 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 ...
6
votes
0answers
1k 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 $95\%$. ...
6
votes
0answers
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 ...
5
votes
1answer
73 views

Functions of continuous random variables

Let Y be an exponential random variable with parameter $\tau > 0$. Compute the cdf and pdf of $F_W$ where $W = Y^3$ The solution states the cdf as $1 - e^{\frac{-y^\frac{1}{3}}{t}}$ because $F_Y =...
5
votes
0answers
52 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
votes
0answers
212 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 ...
5
votes
0answers
545 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 ...
5
votes
0answers
253 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 ...
5
votes
0answers
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
votes
0answers
219 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 ...
5
votes
1answer
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
votes
0answers
171 views

Cox's Theorem: ignorance, objective priors, and the Mind Projection Fallacy

I've been trying to understand Cox's Theorem and the problems surrounding it. There's so much information on this topic that I've become confused as to the exact state of the theorem. I've gathered ...
5
votes
1answer
137 views

How to do inference over two steps in a graphical model simultaneously?

I have observed data $D$ about a physical object described by $M$. I would like to determine the posterior distribution of $M$ given $D$, or $p(M|D)$. Now I can't infer this directly because unknown ...
5
votes
0answers
110 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 $F_n(x)=\frac{1}{n}\sum_{i=1}^n1_{\{X_i\leq x\}}$...
5
votes
0answers
526 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 ...
5
votes
0answers
292 views

Dynamics of birth-death process with discouraged arrivals (alternatively, M/M/1 queue with balking customers)

Take a continuous-time birth-death process, where $k \in \{0,1,\ldots\}$ is the count and the arrival rate of death is $\mu \geq 0$ for $k = 1, 2, \ldots$ the arrival rate of births is $\alpha_k > ...
5
votes
0answers
982 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
votes
0answers
230 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
votes
0answers
183 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
votes
0answers
137 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
votes
0answers
132 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 $\frac{b}{b+w}$ ...
5
votes
0answers
655 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: ...
5
votes
1answer
388 views

Semicolon in probability expression

I run in to this formula when reading a tutorial: $$ \begin{align} P(\pi|\mathbf L;\gamma_{\pi1}, \gamma_{\pi0}) & =P(\mathbf L|\pi)P(\pi|\gamma_{\pi1},\gamma_{\pi0})\tag{28} \\ &\propto [\pi^...
4
votes
0answers
76 views

How to model probabilistic inputs with continuous output using regression

I have trained a multi-output classifier that takes an image as input and returns softmax logits as output. To be specific, the multi-output classifier takes an image and says the probability that ...
4
votes
0answers
47 views

A consistent estimator with infinite expectation?

Typical (or common) approaches to prove an estimator is consistent require finite mean and variance. The proofs usually follow from concentration bounds, e.g. Markov, Chebyshev, etc. I'm wondering ...
4
votes
1answer
70 views

Convergence of scaled $L_1$ distance between two sorted random vectors with same limiting distribution

Let $ X=(X_1,\dots,X_n) $ and $ Y=(Y_1,\dots,Y_n) $ where the RVs $ X_1,\dots X_n, Y_1,\dots Y_n $ are independent and have the same limiting distribution (assume for simplicity that all moments exist)...
4
votes
1answer
58 views

Practical but pretty basic “real world” statistics problem

It was suggested that this forum may be quite helpful. My most humble apologies if I am somehow not posting in the right place! My bosses asked me (I am not a statistician) how many letters have to ...
4
votes
0answers
70 views

Correlated Bernoulli Trials

Suppose there are $n$ dependent Bernoulli trials, $X_{1}$,...,$X_{n}$ with $% X_{j}\in \{1,0\}$ and $\Pr (X_{j}=1)=q$ for all $j=1,...,n$. For any $% n\geqslant 2$ dependent Bernoulli trials, in the ...
4
votes
0answers
40 views

Beta-binomial distribution for scaled and translated Beta

Recall, that a binomial distribution in which the probability of success at each trial is randomly drawn from a beta distribution results in the so called beta-binomial distribution. One can calculate ...
4
votes
1answer
105 views

Intuition behind gradient of expected value and logarithm of probabilities

I recently came across the following curious identity: $$\nabla_\theta \mathbb{E}_{x \sim D_\theta}[f(x)] = \mathbb{E}_{x \sim D_\theta} [ \nabla_\theta \log(D_\theta(x)) f(x)],$$ where $D_\theta$ ...
4
votes
0answers
34 views

Looking for advice: Short-term forecasting using actual forecasts and real time data

First of all apologies, I have very little experience in statistics and my biggest problem is using the correct terminology. I'm here mainly looking for guidance and direction. Background: I have a ...
4
votes
0answers
36 views

Identity on expectation of the minimum of two iid random variables with bounded support

I am reading the 2008 annals of statistics paper "Ranking and empirical minimisation of U-statistics" by Clémençon et. al, and read a statement which I do not know why is true. In order to accurately ...
4
votes
0answers
65 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 ...
4
votes
0answers
299 views

Showing that a Gamma distribution converges to a Normal distribution

Consider $G = \operatorname{Gamma}(p)$. As $p$ goes to $\infty$, the Gamma becomes more and more bell-shaped. How do I show that $\frac{G - p}{\sqrt{p}} \to Z \sim N(0,1)$ as $p \to \infty$? I ...
4
votes
0answers
66 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 ...