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Questions tagged [probability]

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

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Is the sampling distribution of a complete sufficient statistic free from relevant subsets?

Let $T_{\theta}(\mathbf{x})$ be a complete, sufficient statistic $T_{\theta}: \Omega \mapsto \mathbb{R}$, where $T_{\theta}$ is indexed by the parameter $\theta \in \mathbb{R}^n$. Is it true that the ...
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Resources for prerequisites to Probablistic Machine Learning Models

I am a self-learner and have done several machine learning courses but diving into Bayesian or Probabilistic Graphical Models I feel like my prior knowledge is inadequate. I have done some Probability ...
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Multi-dimensional CDF on a discrete support

Suppose I have two discrete-support random variables, $X$ and $Y$. They have joint CDF $F(X,Y)$. If I want to find $\Pr(a \leq X \leq b , c \leq Y \leq d)$. It is obviously not: $F(b ,d)-F(a-1 ,...
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38 views

Computation within log space

What is the conversion of the following equation into log space? $bf2 = 1 + (p * (bf1 - 1))$ Given log.bf1 (log Bayes factor), how do I get to log.bf2 without having to compute bf1, but instead ...
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Discrete Stochastic Processes examples

I am trying to understand various types of stochastic processes. In order for that to happen, I needed some simple examples to be built so that I can build an intuition about them. According to the ...
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Some simple examples to understand various types of Stochastic Processes

I am trying to understand various types of stochastic processes through some analogical examples using simple experiments like a coin toss or die roll. The book of Hwei Hsu (Chapter-5, Page-162-165, "...
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How do you multiply two conditional probabilities?

I have to multiply this: $P(a|b,c)·P(b|c)$ How do you multiply those two expressions? It seems that $P(a|b,c)·P(b|c) = P(a,b|c)$ but I don't know how to obtain that expression multiplying the two ...
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Fitting hedge fund returns to theoretical probability distributions

I know that a lot of work has been done characterizing the first four moments of monthly hedge fund returns across a variety of fund types and strategies, and that work indicates that the higher ...
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1answer
59 views

Genomics Stats Problem

I have used a python script to identify target sequences in a DNA sequence file. There are two classes of sequence: coding and non-coding. I have identified 728 sequences of interest. 597 of these ...
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Probability of deviation of percentages

In a set of N individuals X% have some characteristic. In a random subset of M individuals of this set, we observe that Y% have the characteristic. Note that M is not necessarily very small compared ...
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Using the Brier Score with ordinal data [duplicate]

I have a model built to predict an ordinal response with 3 levels (win, draw or lose, say) and I would like to evaluate predictive accuracy. Is it appropriate to use the Brier Score: $$ \sum_{i=1}^{...
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1answer
45 views

Expected value of ratio of two function of the same random variable

Let $X$ be a r.v. with absolutely continuous distribution and continuous strictly positive density $f: \mathbb{R} \rightarrow [0, \infty)$ and let $g$ a further given continuous density function. Set ...
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Importance of the right-continuity of filtration in definition of strong Markov Property

Taking the definition from wikipedia, With $X = (X_t : t \geq 0) $ as a stochastic process on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with natural filtration $\{ \mathcal{F}(t) \}_{t \...
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What is the probability that the drug synthesized is effective

An experimental protocol developed by NuGenCanPharm Inc to test if a cancer drug is effective is correct 99% of the time, on both effective and ineffective drugs. NuGenCanPharm Inc synthesizes 10,000 ...
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1answer
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Probability of complement events over time

The probability of catching fish in one hour is 0.64? What is the probability of catching fish in half an hour? Regular solution follows the principle for an opposite event: a probability of NOT ...
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34 views

Calibrating probabilities of a binary classifier when class prior is unknown

Is it possible to calibrate the probabilities of a binary classifier when the class priors are unknown? In cases where the data is obtained with selection bias (i.e. more positives than negatives in ...
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1answer
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How to know if a conditional independence is true or false [closed]

I'm learning Bayesian networks and I have to "guess" if the following conditional independence are true or false using the following table: And the conditional independence are: $I_p(A, B)$ $I_p(A, ...
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How to find complete log likelihood for mixture of PPCA

In Appendix C of a paper by Michael E. Tipping and Christopher M. Bishop about mixture models for probabilistic PCA, the probability of a single data vector $\mathbf{t}$ is expressed as a mixture of ...
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1answer
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Choose one of two normal distribution that will give the probability of biggest value when sampling it

Suppose you have two (or more) normal distributions with different mean and variance. You can draw only one sample of only one of the available distributions. Your goal is to get the biggest value ...
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Why is it necessary to assume that a sample vector consists of n sample variables instead of assuming that we have a sample of size n?

This is a very basic question, but I need help to grasp the concept. From what I understand: If I carry out a survey on body weight, every single answer to the question "How much do you weigh?" is a ...
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1answer
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nonexistence of the expected value of heavy-tails distributions

I would be much grateful if you could help me with this. Is the nonexistence of the expected values of Cauchy distribution because of the heavy tails? How can we prove this using calculus or even ...
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Conflicting “facts” about the likelihood employed in Bayes theorem [duplicate]

Consider the following "facts" about Bayes theorem and likelihood: Bayes theorem, written generically as $P(A|B) = \frac{ P(B|A) P(A) }{ P(B) }$ involves conditional and marginal probabilities. ...
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What does it mean to say that $X_1, X_2$ have a “common” Normal distribution?

An exercise question asks Let $X_1, X_2$ be rvs having a common Normal distribution $N(0,1)$ with $\operatorname{Corr}(X_1, X_2) = \rho$. Calculate the coefficient of upper tail-dependence for all $...
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Difference - Probability-To-Exceed (PTE) and $\chi^2$ distribution

I would like to understand the difference between the $\chi^{2}$ distribution and the Probability-To-Exceed ? I have to compare 2 data sets A and B and in the article I am reading, they talk about ...
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Why do we care about the joint distribution of the endogenous variables of a causal model?

In general, we can calculate the joint distribution of the endogenous variables of a structural casual model (SCM) as follows $$ P(X, X_1, \dots, X_n) = \prod_i = P(X_i \mid \text{parent}(X_i)) $$ ...
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Simplification using Cauchy Schwartz Inequality

Can someone please help me understand how the last step in the highlighted part in the equation is arrived at? I did not get how the Cauchy Schwartz Inequality comes into play here. The full paper is ...
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Setting Average posterior probability value in Stata traj plugin

In complement to Aikake Information Criterion, I want to use posterior probability to select the best model for group-based trajectory modeling. In this paper: https://drc.bmj.com/content/bmjdrc/4/1/...
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Coded Caching- comparing diffrent rates

Im trying to understand this inequality but cant understand why it is always true. Its a part of proof which I understand most of it. R* is the oprimtal rate of transmitting when there is average ...
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2answers
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What is P(A|B) in Venn diagram

I am trying to understand $P(A|B)$ in practice. I know that $(A,B)$ is the intersection between $A$ and $B$ on the diagram, and I know as well that $P(A|B)$ is probability of $A$ given $B$. But where ...
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2answers
240 views

How to derive the form of the posterior for regression?

I have seen the general form of posterior for a regression $y = f(x)$ defined as $$ P(\theta|y,x) = \frac{ P(y | x, \theta) P(\theta) }{ P(y|x) } $$ I would like to know how to arrive at this form, ...
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In a test that has multiple choice questions with four options each, when should a test-taker not pick options randomly?

Let's say that a test-taker takes a test with multiple choice questions with four options. Picking the right answer 1 mark and there is a penalty of 0.25 marks (say) for picking the wrong answer. That ...
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2answers
59 views

What would p(a,b|a) be equal to?

What is p(a,b|a) equal to in conditional probability? Any sort of breakdown as to why it might be equal to p(b|a) would be helpful (if it is true in all cases). My reasoning for asking this question ...
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What test to be used for comparing pre and post test results with categorical data?

In the past month(s) I had to get familiar with statistics in order to validate my results although my background is computer science/programming. I currently examine whether I can enhance user's ...
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What is the probability that the new commitment will be fulfilled?

A consulting firm was hired to develop an Engineering project. Based on their previous experience, the direction of this office knows that the time (in months) needed to perform this type of task ...
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1answer
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Probability of choosing a subset of 3 out of 6 with four choices and replacement

With a sack of six marbles labeled 1 - 6 and four picks from the sack (with replacement after each pick), what is the probability that the set of marbles picked will include the subset ...
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1answer
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Why is the probability of a random walk reaching 1 (in n steps) squared greater than the probability of it reaching 2 (in n steps)?

Let $S_n$ be a simple random walk. i.e. $$ S_n = \sum_{t=1}^n X_t, $$ where ${X_t}$ are i.i.d random variables with $$ X_t = \begin{cases} +1, & \textrm{w/ probability } p \\ -1, & \...
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2answers
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Deriving a distribution whose pdf has the shape of a square + a triangle (a right trapezoid)

I want to the derive the PDF which looks like the sum of a triangular and uniform distribution which looks like this: To do this I have simply added the PDFs for the rectangular and triangular parts, ...
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Should I use a binomial glm, beta regression, uniform continuous distribution or nls with logistic regression for probabilities?

I made an individual based model in which there is a population of three types of individuals (lizards) that disperse randomly until they encounter another individual that has a beneficial effect on ...
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1answer
102 views

Time component in the distribution of N elements randomly falling in M bins

I have a system where N elements randomly fall into M bins, so that each bin contains X elements on average. I assume that the distribution of bin occupancies is Poisson(lambda = X). I want to ...
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1answer
16 views

Mutually exclusive events, pairwise mutually exclusive events and disjoint sets

I have been confused by the phrases: mutually exclusive events, pairwise mutually exclusive events and disjoint sets. My book uses these at different places. What is the difference between them?
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1answer
108 views

Is this formula related to the energy function / boltzmann distribution?

I am reading research done in this thesis. On page 20-21 (section 2.1) of the thesis, it describes the following stochastic model: ... let us focus for the rest of this chapter on time-varying ...
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How do you measure the variation of probabilities over time?

Suppose we have $K$ possible outcomes, and a model that assigns a probability to each of those $K$ outcomes over time $t$. That is, suppose for each time period $t$, we have a set of probabilities $(...
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1answer
55 views

convergence in distribution?

I have a question. Let $X_n$ converge to $X$ in distribution, on the other hand, $Y_n$ converges to $Y$. What can we obtain about convergence of division of $X_n/Y_n$ in distribution? Does it ...
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Are all classification models appropriate for estimation of probabilities?

I try to estimate the probability that a tennis player will win based on several predictors (such as skill, form, surface, weather etc.). Can I use every classification method to estimate a ...
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Intuition behind the no convergence of the variance of sum of random variables

$$Var[\bar{X}] = \sigma^2/n $$ $$Var [\sum{X}_i] = n\sigma^2$$ $$lim_{n \to \infty} Var[\bar{X}] = 0 $$ wich means at $\infty$ we will always get the same $\bar{X}$ after every simulation. I ...
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Calculate $P(X>10)$ where $X$ have Poisson distribution

Calculate $P(X>10)$ where $X$ have Poisson distribution $Poisson(7,2)$. using $R$ My attempt By the theory of probability we know $P(X>10)=1-P(X\leq 10)$ Then using R, the probabilitity is: <...
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11 views

Difference between uniform distributed random variable and standard normal random variable [closed]

What are uniform random variables and standard normal variables? What is the difference between them?
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2answers
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Permutation or Combination?

I am a bit confused, if I have 2 football teams, and the results are three (1: home win, 2: away win and X: draw), I can see that the possibilities for these teams results are 9, which formula (...
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1answer
127 views

Binomial CDF - Wikipedia vs Reality

The Binomial distribution article on Wikipedia defines the binomial CDF as $F_{Bin(n,p)}(k) = I_{1-p}(n-k, k+1)$, where $I$ is the regularized incomplete beta function. There is a proof for the ...
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1answer
44 views

Calculating Conditional Expectations

Say $X$ is continuous and distributed according to CDF $G$. How would we calculate $\text{E}[X|X<a]$?