# Questions tagged [probability]

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

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### Probabilities arising from permutations

Certain interesting probability functions can arise from permutations. For example, permutations that are sorted or permutations that form a cycle. Inspired by the so-called von Neumann schema given ...
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### How many sides and what are the probabilities of each side for an unfair die?

Problem I would like to run an inference that predicts from a series of die rolls, how many sides the die has, and what is the probability of landing on each side. Example For example, imagine a die ...
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### How to explain the difference between confidence and credible interval?

The key difference between Bayesian statistical inference and frequentist statistical methods concerns the nature of the unknown parameters that you are trying to estimate. In the frequentist ...
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### 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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### Cox's Theorem: the necessity of (un)countably additivity

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 ...
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### 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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### Distribution that doesn't belong to any maximum domain of attraction?

Question Does there exist a (non-degenerate) distribution that does NOT belong to any maximum domain of attraction? That is: Does there exist any non-degenerate probability distribution function $F$ ...
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We are told that Shaquille O'neal's free throw percentage is 53%. We suppose he shoots a round of 30 and makes 18 of them. a) What is the likelihood that he made the first free throw? My response: ...
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### How to win this dice probability game?

The game is a variation of Pig. Here is how the game works: There are about 20 players. Each round, a single six sided die is rolled. All players add that rolled number to their "bank." However, if a ...
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### How were statistical distributions discovered?

Let me start, that i know that it's not very difficult to generate a probability distribution. If one takes any positive integrable function and normalizes it, this results in a probability density. ...
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### Covariance of order statistics convergence?

Suppose I have a sample $(X_1 \dots X_n)$ and $(Y_1 \dots Y_n)$, all of which are $N(0,1)$ random variables. I am interested in the asymptotic behaviour of $$\frac{1}{n} \sum_{i=0}^n X_{(i)}Y_{(i)}$$...
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### 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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### 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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### Expected number of multinomial samples to cover a multiset

Consider a multinomial distribution $[p_1, \ldots, p_n]$ and a collection of counts $[a_1, \ldots, a_n]$. I would like to know the expected number of multinomial samples needed until every element $i$ ...
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### Summing predicted probabilities from logistic regression using 'one vs. rest'

I have a multiclass classification problem that I have solved using a 'one vs. rest' approach via binary logistic regression classifiers from Python's scikit-learn package. In my problem, there are 3 ...
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### 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 ...