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Questions tagged [conditional-probability]

The probability that an event A will occur, when another event B is known to occur or to have occurred. It is commonly denoted by P(A|B).

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Questions relating the definition of conditional expectation $E[g(X)|M]$ where is $X$ is random variable and M is an event

I saw the following definition of conditional expectation from a book: if M is event and X is continuous random variable then we define: $$E[X|M]=\int_{-\infty}^\infty xf(x|M)dx$$ Which is the ...
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Is this sentence referring to joint or conditional probability?

The following is a quote from my textbook, in a chapter discussing the Viterbi algorithm (Durbin, Richard, et al. Biological sequence analysis: probabilistic models of proteins and nucleic acids. 1st ...
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Estimating incidence on data with limited observation

I have a data of many individuals and wish to know whether a certain event occurred during a specific time frame with each one. The problem is that each individual was observed only a certain portion ...
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Basic probability theory

I was recently given the following statistics: On a particular highway, 18% of drivers are black, 63% of drivers searched by the police are black. So, a black driver is 7.7 times more likely to be ...
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Estimate a probability distribution of target values using features

In my particular problem, I have $$t \in \{1,...N\}$$ time periods, and feature vectors $$x_t \in R^m $$ which I hypothesize predict something about the probability distribution that the targets $...
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How to swap variables in a conditional normal distribution?

I assume that I have two normal distributed variables where one depends on the other: $P(A) \sim N(0,\sigma_a)$ $P(B|A) \sim N(q\cdot A, \sigma_b)$ How can I get the reverse $P(A|B)$ assuming that ...
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Conditional KL Divergence in Clustering Paper

I am trying to implement the following paper on Self-taught Clustering https://www.cse.ust.hk/~qyang/Docs/2008/dwyakicml.pdf. I have the following three co-clustering functions: where p̃(Z|x̃) is Z ...
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Marginal from conditional given no parameter reliance

If $Y|X \sim \text{Normal}(0,1)$ is it true that $Y \sim \text{Normal}(0,1)$. This intuitively seems true as the Normal is characterized by the mean and variance, which have no reliance on $X$. So no ...
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Calculating a probability based on a joint distribution between a Uniform random variable nested within a Uniform(0,1) random variable

Let $X_1 \sim Uniform(0,1)$, and $X_2 \sim Uniform(0, x_1)$, where $x_1$ is the realized value of $X_1$. Find $P(X_1 + X_2 \geq 1)$. I know that I need the joint distribution of $X_1$ and $X_2$. $...
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compared the whole population with the conditional probability

I would like to know how can I compare the whole population with the conditional probability. As the conditional probability is ...
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Ad classification problem, several binary vs. one multiclass model

I have a website that shows 3 ad panels at a time. These ads can belong to three different classes (each ad can only belong to a single class, there is not multilabeling). I want to build a model ...
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Conditional independence of two events

1) If events A and B are independent on given condition C, then does it implies that those two events A and B are independent without the condition C? 2) If events A and B are independent events, ...
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How to create a distribution and sample?

Suppose we are given some small set of data on bundles of electrical wires and increasing voltages run through them, and we note how many of the individual wires fail. So for example, a large data ...
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Finding joint probability distributions from marginal distributions

Question: I was solving test papers where I found this one. My doubt: I know to work with conditional probabilities and Jaccobian Transformation and part A and B can be done applying the above..But ...
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Doubt about state of Predictor Variable in PRF and its implications [Regression Question Series - Part 1]

Given a population $(X,Y)$ we hypothesize underlying population hasa regression line as follows. The conditional expectation is $$\begin{aligned} & E(Y|x) = \beta_0 + \beta_1x & \...
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The likelihood function: Why is it no pdf? [duplicate]

I know that there have already been a lot of questions about why the likelihood is no probability density function and I ve read most of the answers. However, to me the point is still not clear yet ...
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Estimating conditional probability with many samples

I am confused about the estimation of conditional probabilities. Suppose I want to predict a binary outcome variable $Y = 0,1$ given $n$ categorical features $X = (X_1, \ldots, X_n)$, i.e. to ...
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Conditional Probability and its correctness

I am watching this YouTube video on conditional probability example by a university professor. He gives 2 examples: What is the probability of $2$ children being girls if we were told at least one of ...
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Base Rate fallacy in Conditional Probability P(A|B) vs P(B|A)

From Wikipedia: P(A|B) (the conditional probability of A given B) is not equal to P(B|A). For example, if a person has dengue they might have a 90% chance of testing positive for dengue. In ...
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Dependent vs Independent events (Conditional Probability using a pair of dice)

I am watching this video on YouTube to understand dependent and independent events and conditional probability. The example author uses is of a pair of dice. There are 2 dice, Red and Green, let's ...
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how to calculate the conditional probability distribution using N-dimension vectors

I'm reading paper "Recurrent Autoregressive Networks for Online Multi-Object Tracking" In that paper, author mentioned as below image So, X is a N-dimentional ...
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Sampling a conditional joint distribution of continuous random variables using samples from joint distribution and marginal distributions

I am seeking an approach to sampling conditional joint distribution (new to probability). I will put my case in a simple way: Similar question for discrete variables is asked Here but not yet ...
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Likelihood of a linear model in matrix form

I have difficulty finding the likelihood of the data represented in the matrix form. The mapping between target variable $\mathbf{T}$ and observed variable $\mathbf{X}$ is given as $f:\mathbf{X}\...
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I think there is a little mistake in this exercise about the memoryless of Geometric Distribution

An exercise of Jacod and Protter: Let $X$ be Geometric. Show that for $i, j > 0$, $$P(X > i + j | X > i) = P(X > j)$$ I did it and I got a different asnwer: $$P(X > i + j | X > i) = ...
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synthetic datasets without any dependecies between features

I have to create a synthetic data set without any dependencies between features so that this equation should be hold. I thought about to take simply several Gaussians or randomisers, each would be ...
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Probability of an event with multiple conditions

Could you inform me please, how can I calculate conditioned probability of several events? I have 3 events A, B, C; I know P(B|C) and I want calculate P(A|B,C). Is it possible? In the special case ...
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Find conditional probability of multivariate normal

Given vector $X \sim N_2((1,2)^T;\begin{bmatrix} 1/2&&3/2 \\ 3/2&&1/2 \end{bmatrix}))$ find conditional probability of $A={(x_1,x_2):x_1^2+x_2^2=3)}$ given that $x_1=2$ So what ...
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P(X=x|Z=z) given Z=X+Y are all rv's

Let $X,Y,Z$ be random variables where $Z=X+Y$ and $X,Y$ are independent. By Bayes' Law, $$ \begin{align} P(X=x|Z=z) &= \dfrac{P(Z=z|X=x)\ P(X=x)}{P(Z=z)}\\ &= \dfrac{P(Y=z-x)\ P(X=x)}{P(Z=z)} ...
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Deriving marginal pdf from joint pdf

Problem setup: $X \sim \Gamma(\alpha,\beta)$, $f_{Y|X}(y|x)= \tau xy^{\tau-1}e^{-xy^\tau}$ for $y>0$ and $f_{Y|X}(y|x)=0$ for $y\leq0$, where $\tau\geq1$ is a constant. I am asked to derive the ...
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conditional distribution regarding f(x,y,z)

Given that $$f(x,y,z) \propto x^2yz(1-2x-y-z),\mbox{ for }x>0,y>0,z>0, 2x+y+z < 1. $$ Find the conditional distribution of $X\mid Y,Z$. So how I have approached this qns is by using the ...
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Why Does the $\propto$ Symbol Replace the $=$ Symbol When Using Bayes' Rule to Convert Posterior Density to Unnormalised Posterior Density?

My textbook says the following: In order to make probability statements about $\theta$ given $y$, we must begin with a model providing a joint probability distribution for $\theta$ and $y$. The ...
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1answer
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A confusion about Bayes's theorem

I am reading a paper on the differences between bayesian outlook and frequentist outlook. The exact pic from the paper is: I have read a decent amount about what likelihood is and how it is not a ...
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Posterior mean computation of “Monty Hall Poblem”

Background As I understand that the "Monty Hall Problem" is well studied, e.g., here or here, etc. [I am relatively new to probability theory. So, please help me to learn something from you experts ...
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If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$

Question If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$. Attempt: Please check if the below is correct. Let ...
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Conditional probability and independence

Suppose A and Y are discrete dichotomous variables $(A=0,1; Y=0,1)$ If $Pr[Y=1|A=1] = Pr[Y=1|A=0]$, why can we conclude that $$Pr[Y=1|A=1] = Pr[Y=1|A=0] = Pr[Y = 1],$$ without knowing beforehand ...
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Bayesian Statistical Conclusions: We Implicitly Condition On the Known Values of Any Covariates, $x$?

My Bayesian data analysis textbook says the following: Bayesian statistical conclusions about a parameter $\theta$, or unobserved data $\tilde{y}$, are made in terms of probability statements. ...
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How to obtain the posterior distribution of a given problem?

Problem: Compute the conditional distribution of a random variable $X$ given $Y$. If a random variable $X$ is Bernoulli distributed with probability $q$ for $X = 0$. The conditional ...
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1answer
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Binomial probabilities, dividing by probabilities

First, my apologies if this is redundant or out of place. I Googled and searched but found nothing related. I am trying to understand the logic of this equation: $\frac{{p\left( {{\rm{at.least.one....
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Bayesian statistics: probability of next point

I am reading the Deep Learning book and having some difficulties with the following formula (page 134): $$ p(X^{m+1} | x^1, \dots, x^m) = \int p(X^{m+1} | \theta) p(\theta | x^1, \dots, x^m) d\theta. ...
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How to sample for conditional probability from unknown populations

I am providing the full question as well my solution below. I'm looking for help with part (d), a simulation question. Q - Suppose there are two species of Pandas, $T_1$ and $T_2$ which are ...
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How do I adjust action outputs when accounting for volume at which said actions are played?

I apologize for the vague and potentially misleading title, I am very new to statistics and do not yet have a handle on the jargon. Essentially, I have the table below:   ...
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What is the Bayesian Prior Predictive distribution from two normal populations?

The question goes as follows: A shoe factory produces brown shoes and black shoes. They look the same but differ only in their weight characteristics. Brown shoes have their weight distributed as ...
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Probability conditional on a parameter?

This is a definition of the sufficient statistic from Wikipedia. A statistic $t = T(X)$ is sufficient for underlying parameter $θ$ precisely if the conditional probability distribution of the data $...
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Definition of distribution conditioned on both a categorical and Dirichlet prior

If we have a conditional categorical distribution, with unknown parameters, we can represent with a table, as in the example below: \begin{align*} &z \quad P(z|\theta)\\ &0 \quad \theta_0\\ &...
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How parameters formulated for Simple Regression Model

I am reading Simple Regression Model from this book, Section 6.5 (page 267 in downloaded pdf, 276 if viewed online). The author starts with below equation for a simple linear regression model, $$...
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how to generate random integer in a range

If I am given an uniform random integer generator function between range [0, 4] (inclusive), how to design another uniform random integer generation function between range [0, 6] (also inclusive)? ...
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Joint pdf from joint cdf in R

I have a big matrix of data where for each element (column) I have a certain number of values. Using this data I computed, using the Emcdf library the Empirical Joint CDF for every couple of elements. ...
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Distribution of *conditional* frequencies when frequencies follow a Dirichlet distribution

Context: we have a large number of individuals characterized by two binary traits; call these $T$ with values $\{0,1\}$, and $T'$ with values $\{0',1'\}$. So there are four types of individuals: $00'$,...
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Is $P(X+Y>1|X=k)$ equal to $\frac{P(Y+k>1)}{P(X=k)}$?

Suppose there are two random variables $X$ and $Y$. In general does the following hold? Is $P(X+Y>1|X=k)$=$\frac{P(Y+k>1)}{P(X=k)}$? In other words, can I substitute the random variable with ...