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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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Minimal Sufficiency for Two Parameters [on hold]

Let $X_1, ..., X_n$ be iid $f(x; \theta, \lambda) = \dfrac{\lambda e^{-\lambda x}}{1-e^{-\lambda \theta}}$ for x $\in [0, \theta]$. I am asked to find I want to find A minimally sufficient ...
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Meaning of “was produced by the null hypothesis”

There was a question about p value interpretation and one of the comments made me wonder whether there is some basic misunderstanding I have about p values. I wrote "you can discard the null ...
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How to find the conditional CDF based on observed data in R [on hold]

If we have two samples (generally their distribution is not known),say $X\sim N(0,1)$, $Y|X\sim N(X,X^2/2)$. Can we recover the conditional CDF of $Y|X$ based on the observed samples in R? ...
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Probabilistic interpretation of linear regression: Why p(y|x) equal to size of the error?

I was reading Andrew Ng's CS229 lecture notes (page 12). The target function y(i) can be written as: y(i)=θTx(i)+ϵ(i) where, e(i) is the error term that captures unmodeled effects and random ...
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Seeking guidance on determining (predicting) probability (distribution) using Mixture Density (Neural) Networks

I work for an education PAAS (platform as a service) company helping colleges & training institutes manage their course evaluation and recruitment. Students can re-take tests multiple times, tests ...
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identifying which of $d$ normal distribution generated a given sample

I have $d$ Normal Distributions, $N_1(\mu_1, \sigma_1^2) \cdots N_d(\mu_d, \sigma_d^2)$. We pick one of the $d$ distributions with each distribution having a probability of $\frac{1}{d}$ of being ...
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Couldn't understand conditional distribution and density function

The random variable X has a distribution function as shown in the graph below and $ Y=X^2$. Find a. $P(1/2 ≤ X ≤ 3/2)$ b. $P(1/2 ≤ X ≤ 3/2 | Y ≤ 1)$ g. $P(X+Y ≤ 3/4)$ I couldn't understand how ...
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If 'B is more likely given A', then 'A is more likely given B'

I am trying to get a clearer intuition behind: "If $A$ makes $B$ more likely then $B$ makes $A$ more likely" i.e Let $n(S)$ denote the size of the space in which $A$ and $B$ are, then Claim: $...
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Multivariate Theory: How does the new mean only depend on the conditioned variable?

I'm doing some review of Gaussian Processes and Multivariate Normal Theory. I found a really helpful website here, but I have run into a snag. What does the author mean in the sentence below this ...
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How to set up an event with probability of one seventh from coin tossing?

I have been recently asked in an interview a question that: How to set up an event with probability of one seventh from coin tossing? I couldn't find the correct answer and still don't know the ...
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“substituting” random variables in conditional probability?

In a conditional probability distribution $P(A|B)$, can I substitute a new random variable $C$ to represent "A given B" and write it as $P(C) = P(A|B)$? If so, would this expression be true? $$P(C|D)...
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conditional probability: survivial probability of patient in stage 2 given patient survived in stage 1?

This is a basic question, I am however not able to reduce my solution to the simple form proposed in a book and thus seeking help. A patient can go via two stages in a disease. first stage is ...
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Conditional covariance of multivariate normal tail

Let $X\sim N(\mu,\Sigma)$, $t\in\mathbb{R}$, and $a$ be a non-zero vector of the same dimension as $X$. Define a random vector $Y=X\mathbb{1}(a^\top X\ge t)$, where $\mathbb{1}$ denotes the indicator ...
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Computing posterior based on sum of multivariate normal distribution

Currently I am exploring topics for my undergrad thesis. Although I took a course in Bayesian statistics, I am not yet sure how to proceed in finding the posterior in the following case. I have a d-...
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To find the conditional probability of heads in a coin tossing experiment

Question . A coin box contains 8 fair, standard coins (head and tails) and 1 coin which has heads on both sides. A boy selects a coin randomly and flips it 4 times, getting all heads. If he flips ...
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Drawing two balls from two urns

Assume that there are two urns. The first urn contains 4 red balls, 3 blue balls, and 3 white balls. The second urn contains 2 red balls, 4 blue balls, and 4 white balls. You randomly select an urn ...
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Joint Probability check

There are 6 players and 18 cards. Each of the 18 cards is numbered 1-18 (each is unique). Each player is dealt 3 cards. Players A and B are the first two players in the deal. The deal was a uniform ...
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Probability of Consecutive rainfall days?

I have a daily rainfall data measured in (cm), i have put some threshold and derived a categorical variable whether it is a rainy day or a dry day. From this data i want to calculate Probability of a ...
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conditional pdf and joint pdf

I am looking at a description of a process that says $f(y|a_1,z,a_0) = \dfrac{f(y,a_1,z,a_0)}{p(a_1|z,a_0)p(z|a_0)p(a_0)}$ I am not sure if I follow this joint pdf, conditical pdf , p(.) relation. ...
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Notation: in general is it ok to think of $p(y|x)$ as $p_x(y)$?

In the notation for the conditional probability, $p(y|x)$, I believe usually we think of this with $x$ having a fixed value and $y$ varying, such that it integrates to one. In this case, is it ...
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antagonistic simulated annealing

Simulated annealing aims at a series of target distributions $$\pi_T(x)\propto\exp\{T\,H(x)\}$$ to find the maximum of the function $H$ and its argument $$\arg_x\max_{x\in \mathfrak X} H(x)$$ if the ...
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Bayes Theorem in call center routing problem?

I want to route the call to the agent to maximize customer satisfaction. I have following data: Probability that customer is currently happy p(C+), we have this from sentiment analysis of the reviews/...
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conditional probability involving mixed variable types

I'm trying to answer the following question A defective coin minting machine produces coins whose probability of heads is a random variable $T$ with PDF $f_{T}(p) = 1+\mathrm{sin}(2\pi p)$ if $p \in ...
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Why dependencies would cancel while paramterizing CPD?

Consider that in CPTs of a Bayes Net(structure is fixed) one dependence is fix(parameters are set for $P(X|Y)$) and we are parameterizing other CPDs, is it possible that the other parameters would ...
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Conditional Independence

I have a joint probability, which factors as follows: $P(A,B,C,D) = P(A,B) \cdot P(C|A) \cdot P(D|B)$ So I know that $C$ and $D$ are independent given $P(A, B)$ right? I want to infer $P(A,B|C,D)$....
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Ranking the conditional probability of classification study (PLSDA) to get better accuracy

I am building a model to predict the probability of an animal to be poor or good, using PLSDA. The sensitivity and specificity are 76% and 56%, respectively. Let's ignore the sensitivity for now, ...
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Conditional distribution of $X_1=S_1+T$ and $X_2=S_2+T$

$S_1, S_2$ and $T$ are independently and uniformly distributed on $[0,1]$. What will be the conditional distribution $G(x_1|x_2)$? I have no idea on how to proceed. Any help will be appreciated. ...
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Finding $E(X\mid X>Y)$ when $X,Y$ are i.i.d $U(0,1)$

I am unable to compute conditional probability(x|x>y) in the above question. Also, I am unable to determine the region of integration for calculation of the above expectation.
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importance sampling for conditional probability

Stats experts, I am trying to estimate $G(x)$ for $x$ in a very small subset $A$ of a finite space $U$, i.e $E(G(x | x \in A))$. Determining if $x$ in $A$ has a cost. So it is expensive to uniformly ...
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Expectation of a function of random variables

I'm trying to simplify the following expectation so that I can later solve a maximization problem: $max_k E[(A - kB)^2]$, where $A$ ~ $N(0,\sigma^2_1)$, $B = A+ \epsilon$ and $\epsilon$ ~ $N(0,\sigma^...
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Conditional Probability Based on Sample When Population is Known

I have a count from each U.S. state of people that ride bikes. I want to know whether the count is larger or smaller than expected relevant to the population of each state (I also have the population ...
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Compute true positive rate from only predicted probability and actual probability (Binary LDA classifier)

I'm sort of stuck on this question and I can't find a similar problem online. Consider the following to be given: 1.input x is 1D, output y is binary {0,1} 2.marginal probability of y is $\pi_y=P(y)$ ...
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Fitting Beta Distribution Parameters to Y conditional on X

I have a bi-variate data set where Y is in [0,1]. X is some measure of intensity and in this example is (0,~200) though there is no hard upper bound. X has a strong positive skew but I am not ...
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Condition Probability - What am I doing wrong?

I am reviewing some notes of mine and refreshing myself with some statistics and I came across a problem that asks me to calculate $P(X>1|Y>1)$ for the random variables $X$ and $Y$ whose joint ...
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Discussion on a probability question from Puzzling.SE

BACKSTORY AND RELEVANT LINKS So over on Puzzling.SE, a user asked this question. DISCLAIMER: I'm not asking you guys for the answer to the question, but rather an answer to the discussion that ensued ...
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Calculating Conditional Probability in a small cipher problem

I have a small part of data that is n bit long (0 or 1). Probability that a bit is 0 equals p. I also have a key that is used to cipher this data that is 1 bit long. The key will be 0 or 1 for the ...
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How to approximately sample an unknown non-parametric joint distribution given a complete set of partial conditional distributions?

This question is related somewhat to Bayesian networks. In a BN, you have a DAG (directed acyclic graph). By supplying the root nodes with a sample, you can then follow the directed arcs to sample ...
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Do we assume conditional independence to prove it?

Sorry if this is a silly question but it comes from reading a book on directed graphical models. They show algebraically that two variables $x$, $z$ are conditionally independent given $y$. It says: ...
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Why does the order of events A and B not matter with conditional probability? p(A | B) x p(B) = p(B | A) x p(A)

Please can someone explain why the order of events with conditional probability does not matter? p(A and B) = p(A | B) x p(B) = p(B | A) x p(A)
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Effect of a class to another class's passing rate

I have a case where there are 2 classes and I want to figure out how class 1 effects the passing rate of class 2. An example case: ...
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Prove limiting distribution goes to stationary distribution: $\lim_{t\to\infty} \pi_{j}(t) = \overline{\pi}_{j}$

This is a problem I'm struggling with on continuous-time Markov chains. Here, we are considering a continuous Markov process with phase space $\{1, 2\}$ (there are only two states). Moreover, $\...
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How to decide which Probability distribution to use on a specific problem?

I need some guidance. Which probability distribution I can use to understand the number of orders one person can handle in a company? It's like each person has orders which takes minimum of 50 days to ...
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Can the hazard function be defined on a continuous state

The hazard function is defined as the instantaneous failure rate or instantaneous hazard rate as $\Delta t$ tends to zero $$ h(t)= \lim_{\Delta t \to 0} \frac{R(t)-R(t+\Delta t)}{\Delta t * R(t)} $$ ...
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Probability that feature selection in elastic net regularisation is meaningful - evaluating the statistical significance of chosen features

I have a research question - can I use baseline clinical features to predict my binary clinical outcome in individual patients? I am interested if the performance of my model is greater than chance. I ...
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Is there an error in those tables?

I recently came across this popular answer on Stack Overflow (1st result for profiling code linux on Google). I have some knowledge of Bayesian Statistics and I ...
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What can we say about $N_{i}$ where $N=N_{1}+\cdots+N_{m}$, $N\thicksim Geom(\frac{1-p}{p})$ and conditional distribution of $N_{j}$ is binomial

Suppose that the number of events $N$ is a Geometric random variable with mean $\frac{1-p}{p}$. Further suppose that each event can be classified into one of $m$ types with probabilities $p_{1},p_{2},\...
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How to generate a conditional random variable in R? [closed]

Suppose there is a sample $X\sim N(0,1)$ x<-rnorm(100). If I want to generate a conditional random variable $Y|X\sim U(0,1)$, how can I get this conditional ...
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Why is the conditional distribution from a multinomial a binomial with these parameters?

Say we have a multinational distribution , $$Y=(Y_{1},Y_{2},Y_{3},Y_{4},Y_{5}) \sim multi(n,\frac{1}{2},\frac{\theta}{4},\frac{1-\theta}{4},\frac{1-\theta}{4},\frac{\theta}{4})$$ with observed $X=(...
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What method to use for conditional density querying

I have a dataset of 3d poses each represented by 40 points (all relative to the central point). So my data has dimensionality 120. What is needed is to learn how build realistic pose, when positions ...