Questions tagged [probability]
A probability provides a quantitative description of the likely occurrence of a particular event.
3,453
questions with no upvoted or accepted answers
17
votes
0
answers
2k
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 \...
- 828
12
votes
0
answers
986
views
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 ...
- 348
10
votes
3
answers
192
views
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 ...
- 203
10
votes
1
answer
427
views
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 ...
- 64.2k
9
votes
0
answers
119
views
Intuitively, why do flushes overtake straights as the number of cards in hand increases?
A 5-card poker hand is more likely to be a straight than a flush.
But a 13-card bridge hand is more likely to contain a 5-card flush than a 5-card straight (source: I read it online somewhere).
What ...
- 4,119
9
votes
0
answers
508
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 ...
- 475
8
votes
0
answers
302
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 ...
- 3,358
8
votes
0
answers
2k
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 ...
- 383
8
votes
0
answers
280
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 ...
- 551
8
votes
0
answers
383
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 > ...
- 535
8
votes
0
answers
631
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|$.
$Y=|X|$ ...
- 631
8
votes
0
answers
427
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 $Y_2$ ...
- 101
7
votes
1
answer
166
views
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 $\...
- 129
7
votes
1
answer
722
views
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 ...
- 1,194
7
votes
0
answers
144
views
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 ...
- 71
7
votes
0
answers
703
views
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 ...
7
votes
2
answers
203
views
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 ...
- 71
7
votes
0
answers
2k
views
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 $\mathcal{U\left(a, b\right)}$, $...
- 245
7
votes
1
answer
1k
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 $...
- 4,571
7
votes
0
answers
129
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 ...
- 4,571
7
votes
0
answers
282
views
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 ...
- 551
7
votes
0
answers
2k
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 $95\%$.
...
- 233
7
votes
0
answers
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 ...
- 1,661
7
votes
0
answers
161
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 $\frac{b}{b+w}$ ...
- 555
6
votes
0
answers
172
views
Estimating Coin Flip Probabilities with Missing Information
I am trying to create an example that shows how the quality of estimation is impacted by incomplete information (e.g. deliberately neglecting the Markov Property) and wrong assumptions about the data ...
6
votes
0
answers
125
views
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: ...
- 61
6
votes
0
answers
90
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 ...
- 381
6
votes
0
answers
1k
views
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. ...
6
votes
0
answers
2k
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 ...
- 850
6
votes
0
answers
565
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 ...
- 311
5
votes
0
answers
50
views
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 $i$ ...
- 51
5
votes
1
answer
585
views
Understanding Sequential Model-Based Optimization for machine learning
One of the hyperparameter tunning approaches that I came across recently is Sequential Model-Based Optimization (SMBO), which is a very smart approach that uses previous iterations in order to find ...
- 689
5
votes
0
answers
82
views
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 $(...
- 397
5
votes
1
answer
740
views
What is p(data) in image generation
In the context of image generation architectures such as VAEs or GANs (say we are using mnist digits) what do we mean by probability distribution of the data?
Just to clarify this question and make it ...
- 758
5
votes
0
answers
648
views
What conditions are needed for $a_n = O_p(n^d) \implies E[a_n] = O(n^d)$?
Let $X_n$ be a uniformly integrable sequence of random variables. In a recent question I asked about the possibility of converting Big $O_p$ convergence in probability of the sequence $X_n$ to Big $O$ ...
- 940
5
votes
0
answers
3k
views
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 ...
- 589
5
votes
0
answers
239
views
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}...
- 169
5
votes
0
answers
309
views
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
$$\frac{1}{n} \sum_{i=0}^n X_{(i)}Y_{(i)} $$...
- 51
5
votes
0
answers
3k
views
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, . ....
- 53
5
votes
0
answers
340
views
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 $$\frac{...
- 105
5
votes
0
answers
179
views
Generalization of Dirichlet distribution over matrix
I know there is generalization of normal distribution of matrix-valued random variable, i.e., Matrix normal distribution. I wonder whether there is generalization of Dirichlet distribution that each ...
- 221
5
votes
0
answers
1k
views
Under what conditions will a Bayesian posterior fail to converge to a point mass?
Let's say you have a Bayesian model:
$$\theta' \sim g(\theta|\mu) $$
$$ y \sim p(y|\theta')$$
And we have some data ($n$ data points) $\mathbf{y}_n$, which we will use to perform inference on $\...
user145807
5
votes
0
answers
473
views
Probability with an unbalanced coin where consecutive flips are not independent
Thanks in advance for the help.
Suppose someone has an unbalanced coin that they flip 100,000 or so times in a row. This person then gives you the results. You do not know the probability of ...
- 893
5
votes
0
answers
2k
views
Standard normal quantile approximation
In the book Asymptotic theory of statistics and probability by Anirban DasGupta (2008, Springer Science & Business Media) in page 109 Example 8.13 I found the following approximation
$$\Phi^{-1}\...
- 1,385
5
votes
0
answers
115
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....
- 4,384
5
votes
0
answers
119
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 ...
- 51
5
votes
0
answers
298
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 ...
- 16.7k
5
votes
1
answer
3k
views
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 $\sigma_2$ ...
- 573
5
votes
0
answers
1k
views
Empirical multivariate probability integral transform
Is there a 'simple' way to obtain a non-parametric empirical multivariate probability integral transform?
Univariate case
The probability integral transform relates to the transform of any random ...
- 4,408
5
votes
0
answers
199
views
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 ...
- 343