# Questions tagged [probability]

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

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### 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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### 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,...
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### 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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### 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 $Y_2$ ...
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### 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 ...
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### 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 ...
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### 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)...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $D_\theta$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
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 ...