Questions tagged [probability]
A probability provides a quantitative description of the likely occurrence of a particular event.
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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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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 ...
user75138
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How to guess the size of a set?
Assume we have a set of unique words and draw a number $n$ of them using simple-random-sampling without replacement independently in each round. We have several rounds and try to guess the set size ...
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Queuing theory for elevators
It's been a while since I had my probability course based on Sheldon Ross' book "Probability Models", and while I never went into econometrics, I was very interested in the queuing theory section. I ...
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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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Why we really need the concept of "Local" Rademacher complexity?
Recently, I have been studying High-Dimensional Statistics: A Non-Asymptotic Viewpoint written by Martin J. Wainwright. In this book, the author uses a special complexity measure which is called Local ...
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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 ...
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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 ...
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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 ...
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Difference between pointwise and uniform confidence sets
According to Wasserman's All of Nonparametric Statistics p.6, let $\mathfrak{F}$ be a set of distribution functions $F$ and $\theta$ some quantity of interest. A uniform asymptotic confidence set $C_n$...
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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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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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Posterior consistency for scale-mixture shrinkage priors in low dimension?
Consider the model [1]
$$y_n=X_n\beta_n+\epsilon_n$$
$$\beta_i|\sigma^2,v_i \sim \mathcal{N}(0,\sigma^2 v_i), i=1,\ldots,p$$
$$v_i \sim \beta^\prime(a,b)$$
$$\sigma^2 \sim \mathcal{IG}(c,d)$$
where $\...
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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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Distribution of the $L^2$ norm of a vector of components drawn from uniform distributions
We consider a random vector $\vec{v} = \left(x_{1}, x_{2}, \dots, x_{n}\right)$ built from $n$ real random variables drawn from a real continuous uniform distribution $\mathcal{U\left(a, b\right)}$, $...
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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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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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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 > ...
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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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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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Definition and Interpretation of Likelihood for non-PhD's
Regarding "non-PhD" in the question title, please answer this question for the audience of people with a solid understanding of probability distributions but no knowledge of the nuances of ...
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Assign Numbers to N-sided Die to Make the Expectation Close to the Fair Mean
I got the following question in one interview: suppose we have an N-sided die and given the probability of landing on each side, how to assign values from 1 to N, to make the expected value close to $(...
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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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Am I crazy? I think my stats professor is wrong about this simple probability question
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 ...
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Kolmogorov distributuon derivation
I would like to know if there is a book talking about the derivation of Kolmogorov distribution (Using usual definition for the bridge process)
\begin{align}
P(K\leq x)=1-2\sum_{i=1}^{\infty}(-1)^{i-1}...
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How to correctly re-formulate mutual information between time series?
I am trying to understand the re-formulation of mutual information between time series presented in Galka et al.
They note "If the data is given as a pair of time series $x_t$ and $y_t$, $t = 1, . ....
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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 ...
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Variance of quotient of Poisson random variable and sum of the Poisson sample
Let
$$Y_1\sim \operatorname{Poisson}(\lambda_1)\\Y_2\sim \operatorname{Poisson}(\lambda_2),$$ where $Y_1$ and $Y_2$ are independent, and $\lambda_1, \lambda_2>0$.
What is the variance of $$\frac{...
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Generalization of Dirichlet distribution over matrix
I know there is generalization of normal distribution of matrix-valued random variable, i.e., Matrix normal distribution. I wonder whether there is generalization of Dirichlet distribution that each ...
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Under what conditions will a Bayesian posterior fail to converge to a point mass?
Let's say you have a Bayesian model:
$$\theta' \sim g(\theta|\mu) $$
$$ y \sim p(y|\theta')$$
And we have some data ($n$ data points) $\mathbf{y}_n$, which we will use to perform inference on $\...
user145807
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Probability with an unbalanced coin where consecutive flips are not independent
Thanks in advance for the help.
Suppose someone has an unbalanced coin that they flip 100,000 or so times in a row. This person then gives you the results. You do not know the probability of ...
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Trading signals example from Marginal Revolution blog
From http://marginalrevolution.com/marginalrevolution/2016/09/someone-give-doug-blog.html:
Many trading signals reliably predict prices, but not strongly enough
to overcome transaction costs (i.e....
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How can I calculate the probability that the product of two independent random variables does not exceed $L$?
I have one variable, $X$, which is provided hourly for a period of one month (720 total values in the series). I have another variable, $Y$, which is provided quarterly (for which I am provided the ...
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I'm not asking for a conjugate prior. Is there a distribution $p(x|y)$ that satisfies $\int p(x|y)Beta(y|a,b) dy = Beta(x| a', b')$?
I know the result of integrating a Gaussian against another Gaussian is still Gaussian, $$\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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Probability that one random variable is larger than another with known correlation
Let's say I have a normally distributed random variable $X_1$ with known standard deviation $\sigma_1$ and $E[X_1]$ is $0$. Let's say I have another variable with known standard deviation $\sigma_2$ ...
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How can I determine the expected value / risk of ruin of game where probability changes dependent on state?
Consider the following game:
You are given a random number generator which you can use to play a game. The object of the game is to reach the final tier after which you collect prize tokens. In ...
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Local Version of Bernstein Von-Mises Theorem?
The Bernstein-Von Mises theorem says that, under reasonable conditions, the posterior distribution $p(\theta | x_{1},\ldots,x_{n})$ converges weakly to the normal distribution after suitable rescaling....
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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 ...
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