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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Computing a marginal distribution of a joint involving a delta function

Suppose that we have four continuous random variables $x,y,z,$ and $v$ and we want to compute the following integral: $$\int f(x\mid y)f(z\mid x,y)f(v\mid z,x,y)\,dx$$ There are a few conditions: $...
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9 views

Correct terminology to refer to a collection of conditional probability distributions

I would like your help to find the correct terminology to refer to a collection of conditional probability distributions. Consider two random variables $Y$ and $X$, with supports $\mathcal{Y}$ and $\...
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21 views

Changing a conditional probability to a deterministic function

Suppose that we have a conditional density function $p(y|x;\theta^*)$, where $\theta^*$ represents distribution parameters and are assumed to be deterministic. Is it possible that we write this ...
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47 views

What's the role of the data in Gibbs Sampling?

Trying to wrap my mind around Gibbs Sampling. Across many answers in this same forum, I constantly notice that the examples shown do not actually require an observed data set (First example (with R ...
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Conditional probability of n sequential time series points

I have hourly road traffic volume data. While I assume the volume data can be defined as normally distributed, there is some correlation. i.e. The traffic in the past hour may affect the traffic in ...
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21 views

Conditional coverage probability

Suppose I'm interested in a population mean $\mu$ of some finite discrete variable $Y \sim P$ defined on $\mathbb{R}$. I have a $100(1-\alpha)\%$ confidence interval for the sample estimate $\hat{\mu}$...
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13 views

How do I calculate the standard deviation of a mean value for a joint distribution? [closed]

Say I have a function $f(\mathbf{x})$ where $\mathbf{x}$ is a vector of 0s and 1s. Each of these 0s or 1s, $x_i$, comes from a probability distribution $p(x_i | x_1, \dots, x_{i-1})$. When I sample ...
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Consistent estimator for conditional expectation

Take sequence of random vectors $(Y_i, X_i)_{i=1}^N$ i.i.d. $X_i$ has finite support. Let $x$ be a point in the support of $X_i$. Consider $E(Y_i|X_i=x)$. Suppose it exists and is finite. Is it ...
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Implications of conditional mean independence

I have three random variables $Y,X,W$ with supports $\mathcal{X}, \mathcal{Y},\mathcal{W}$, respectively. I assume $E(Y|X,W)=0$ almost surely. Take two functions $z: \mathcal{X}\rightarrow \mathbb{...
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54 views

$X\perp Y \Leftrightarrow Y\perp r(X)$?

Consider the random variables $Y$ and $X$. Let $\mathcal{X}$ denote the support of $X$. Let $\mathcal{Y}$ denote the support of $Y$. Let $r:\mathcal{X}\rightarrow \mathbb{R}$. I have doubts about the ...
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In Gaussian processes, why does the conditional Gaussian “agree” with data?

I'm learning about GPs, and one thing I don't quite understand is how the posterior works. Consider this figure: Rasmussen and Williams say: Graphically in Figure 2.2 you may think of generating ...
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Obtaining conditional density while sampling from exponential distribution

Given : $X$ and $Y$ are independent exponential random variables with means $\theta$ and $2\theta$ respectively, the conditional density for $(X,Y)\mid X + 2Y $ is desired problem in attempt : I ...
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Troubles with Basket-Sensitive Recommendation System using CF + Modified Random Walk

I'm trying to reproduce and understand the "Basket-Sensitive Random Walk" for recommendation systems proposed in the paper "Grocery Shopping Recommendations Based on Basket-Sensitive Random Walk" of ...
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2answers
80 views

Conditional distribution of arrival times in Poisson process

Suppose I know over a window $[0, T)$ that I have observed $n$ samples from a poisson process $N_t \sim p(n|\lambda t) = \frac{1}{n!}(\lambda t)^{n}\exp(-\lambda t)$. What is the conditional ...
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Estimate run time given widget has run a given time without failure

Given the following distribution of the fraction of widgets running over time, how do I calculate the expected run life of a widget (d_f days) given that it has already run d_c days? My widget ...
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Conditional model with correlation between estimations

I am trying to estimate Click-Through-Rate (CTR) given the following two models: $$P(Click|Visible)$$ $$P(Visible)$$ The output is: $$P(Click) = P(Click|Visible)*P(Visible)$$ Unfortunately, $$P(...
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Application of law of iterated expectations

I would like your help to show a statement that uses the law of iterated expectations. In my notation $Supp_X$ denotes the support of a random variable $X$. Consider the random variables $\epsilon,...
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Deconvolution of the sum of three gaussian distributions

Consider the sum of three normal random variables: $ R_{i,j}=A_{i}+B_{j}+C_{i,j}\, $ where $ A_{i}∼N(μ_{A},σ_{A}) $ , $ B_{j}∼N(μ_{B},σ_{B}) $ and $ C_{i,j}∼N(μ_{C},σ_{C}) $ . Assuming $A$, $B$ ...
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Sample selection bias

In Learning and Evaluating Classifiers under Sample Selection Bias, we suppose that examples $(x, y, s)$ are drawn independently from a distribution $D$ with domain $X × Y × S$ where $X$ is the ...
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Double selection with varying size selection set (beginner)

I'm self-taught in statistics, so I have some holes in my knowledge for sure. please bear with me. I have a hard time defining my problem. I want to figure out if my selection procedure (governed by ...
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Can independent/dependent events be thought of graphically?

My understanding is that events are subsets of the total outcomes in a sample space. So if two events are mutually exclusive, then they (the sets) do not overlap in the sample space. This can be seen ...
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Standard error of conditional survival probability using delta method

I need to estimate the standard error of the conditional survival probability using the delta method. I fitted a kaplan meier curve for these probabilities.I know how the delta method works, I just ...
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sum rule in conditional probability

P and S are the common cause of c. If P(C=true| P,S ) is given , can I introduce S to P(C|P) as P(C= true|P= true)= P(C=true| P =true , S= true)* P(P=true ,S=true )+ P(C= true | P=true ,S =false )*P(P=...
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How to eliminate variable given conditional probabilities

$P$ and $S$ are the common cause of $c$. If $P(C=true| P,S )$ is given as the table below, and $P(S=true) =0.3$, $P(P=true) =0.9$ how can I eliminate $S$ and calculate $P(C=true | P=true )$ and $P(C= ...
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how to multiply two conditional probabilities in general

I am trying to understand how to multiply two conditional probabilities. $P(X|C) \times P(C| P,S)$ seems to equal to $P(X,C | P,S)$. How to understand this? I understand the product rule, but how ...
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21 views

Deriving the conditional distribution of a multivariate normal, for inequalities

This question is slightly related to Deriving the conditional distributions of a multivariate normal distribution. In that question, the following situation was given. If $Y$ follows a multivariate ...
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23 views

Sequence of shifted exponential distributions has uniform conditionals?

Suppose we generate a sequence of random variables as follows. First, we let $X_1 \sim Exp(\lambda)$, where $Exp(\lambda)$ denotes the exponential distribution with rate parameter $\lambda>0$. For $...
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27 views

Name for decomposition of joint probability into product of conditional probabilities

Consider a set of events $A_1, \dots, A_n$. By definition we have $$\mathbb{P}[A_1 \cap (A_2 \cap \dots \cap A_n)] = \mathbb{P} [(A_2 \cap \dots \cap A_n) | A_1] \mathbb{P}(A_1).$$ Applying the ...
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Finite second moments inhertitable to conditional variables?

Assume a random vector $\mathbf{x}=(x_1,\ldots,x_n)^\top$ that has finite second moments, i.e., $$\int\mathbf{x}\mathbf{x}^\top\rho(\mathbf{x})\,\text{d}\mathbf{x} < \infty.$$ Does it follow that ...
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Conditional Probability Table in R

I want to perform Bayesian network analysis in R. I have a large network and i am bit confused with defining conditional probability tables! In my network i have a node with in-degree of centrality ...
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If $X\sim \operatorname{lognormal}$ then $Y:=(X-d\mid x\geq d)$ has approximately a Generalized Pareto distribution

Let $X$ be a random variable with lognormal distribution. Show that when sufficiently large then $Y:=(X-d\mid x\geq d)$ is approximately a random variable with generalized Pareto distribution. Hint: ...
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How does the probability of events change if an event does not occur

Suppose that someone tells me I will collect $\$100$ dollars within some time interval. Those time intervals are 1 to 7 days, 8 to 30 days and eventually after 30 days. Let $A$ be the event I ...
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State value function $v(s)$ as an expectation [duplicate]

Suppose we have a Markov Decision Process with environment states $s \in S$, agent actions $a \in A$, and rewards $r \in \mathbb{R}$.So if an agent takes an action $a_t$ in state $s_t$, he will end up ...
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pdf from a set of conditional pdfs

I have an interesting problem, i have seen in many text books ways of calculating conditional pdfs but not many where given a set of conditional pdfs for a variable we wish to calculate it's pdf. In ...
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35 views

Does the family of binomial distributions conditioned on $X > 0$ belong to the exponential family?

Does the family of distributions where $p(x,\theta)$ is the conditional frequency function of a binomial $\mathcal{B}(n,\theta)$, variable $X$, given that $X > 0$, belong to the exponencial family? ...
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variance of multinomial distribution

Assume $A_{kj} \sim$Multinomial$(1, \;\underbrace{(1/m, 1/m, ..., 1/m)}_{\textrm{m times}})$, where $k=1,2, ... m$ and $j=1,2, ... n$. It is clear to see that $\sum_{k=1}^mA_{kj}=1$. If we impose a ...
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34 views

Showing $Y_n\stackrel{p}\to Z$ where $Y_n=B_nZ+(1-B_n)X$

I am reviewing some of my old class notes again, and I came across the following problem. I think I have solved the problem correctly, but I wanted to see what others here thought. Do you think I ...
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42 views

Derivation of the formula for the probability of a class, given conditionally independent attributes

The following is a formula that finds the posterior probability of a class (i.e. yes or no) given four conditionally independent attributes: $$P(c|X) = P(x_1|c)\cdot P(x_2|c)\cdot P(x_3|c)\cdot P(x_4|...
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66 views

Conditional probabilities for sequence of events

The following table represents all possible paths of dichotomous events at 5 time moments. At each time moment either 1 or -1 event occurs with probabilities $p$ and $q$. Time stops when one observes ...
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1answer
19 views

An trick involving indicator variables, conditional probabilities and expepectation

I am reading a paper on a statistical model for credit risk management. The details of the model are not important, but I mention for context. Suppose that we are interested in a stochastic process ...
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2answers
60 views

What is the number of independent parameters in a family of conditional probabilities?

Suppose we have $n$ events $X_1, X_2, ..., X_n$ and we write down every possible conditional probability we can form from a subset of these events. So we're interested in all probabilities such as: $...
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39 views

Does independence and mutual exclusivity induce impossibility?

Given that we know A and B are independent and they never occur at the same time, one of them must be impossible, no? $$ P(A\mid B)=\frac{P(A \cap B)}{P(B)}\\ \text{if A and B independent, B gives no ...
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38 views

Cumulants of Poisson random variable conditioned on a Bernoulli random variable

Consider a Bernoulli distributed random variable $Y$, which is 1 with probability $p$ and 0 with probability $1-p$. Further there is a random variable $X$ where the conditional probability ...
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1answer
32 views

Confused about conditional counterparts to traditional probability laws

I'm self-studying probability and have seen the following in various readings. The "conditional counterpart" $$P(x,y|\theta) = P(x|y,\theta)P(y|\theta)$$ to the traditional conditional probability ...
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Conditional probability question for healthcare scenario

I'm trying to determine the probability of an event occurring, given that another event has already occurred. Here's the scenario: Surgeons are allocated a block of time to perform cases. For ...
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2answers
25 views

How to write $P(\delta = 1 | X = x)$ as function of $P(\delta = 1 | X = x, Y = y)$

Suppose that $\delta$ is a Bernoulli random variable, and suppose that $X, Y$ are continuous random variables. Is there a way to write $P(\delta = 1 | X = x)$ as function of $P(\delta = 1 | X = x, Y ...
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Calculate $\mathbb{P}[Y=y|X=x]$ where $X$=# claims reported diring firs year, $Y$=# claims that will eventually be reported

A property-casualty insurance company issues automobile policies on a calendar year basis only. Let $X$ be a random variable representing the number of accident claims reported during calendar year ...
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Markov's inequality with conditional probability

How to apply Markov's inequality in case of conditional probability $P(X \ge a | Y \le a)$ where $X$ and $Y$ are not independent. Can we write $P(X \ge a | Y \le a) \le \frac{E(X | Y \le a)}{a}$?
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Conditional transformation of variables

I've seen a trick for finding the p.d.f of $r(X,Y)$ where $X$ and $Y$ are r.v's by first calculating the cdf i.e $P(r(X,Y) \leq l)$ and then differentiating to find the pdf. So if $\Omega = \{(x,y) | ...
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42 views

Compute $E(X_1|X_1+X_2)$ $X_1, X_2$ both iid $Exponential(1)$

I recently stumbled across this question on CV: Conditional expectation conditional on exponential random variable And really liked the answer provided by @Rush, but I wanted to try to compute this ...