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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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How do i solve this problem on conditional probability [closed]

Where x~ u(0,4), what is P(X>Y|X<2Y) I know that X lies between Y at the lower bound and 2Y at the upper bound but i am having trouble trying to figure out P(X<2Y) as none of the examples i ...
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Are the marginals of these two joint distributions the same? [closed]

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and let $X:(\Omega, \mathcal{A})\rightarrow (\mathcal{X}, \mathcal{F})$ and $Y:(\Omega, \mathcal{A})\rightarrow (\mathcal{Y}, \mathcal{G})...
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Birnbaum's Theorem: Strong belief in a model $\implies$ the likelihood function must be used as a data reduction device?

Working through understanding section 6.3.2 (pg. 292-294) in Casella and Berger's Statistical Inference (2nd-ed). The following definitions and principles are given: Definition (Experiment): An ...
Aaron Hendrickson's user avatar
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Equal joint distribution for different conditional-marginal pairs

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and let $X:(\Omega, \mathcal{A})\rightarrow (\mathcal{X}, \mathcal{F})$ and $Y:(\Omega, \mathcal{A})\rightarrow (\mathcal{Y}, \mathcal{G})...
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How to evaluate this conditional expectation for the E-step in expectation-maximisation?

I'm trying to devise an expectation-maximisation algorithm for a certain problem but I'm unable to derive the conditional expectation in the E-step. For the purpose of this question I'll simplify the ...
DangerousTim's user avatar
3 votes
1 answer
109 views

Applying Bayesian probability to a generalized Monty Hall problem

I posted this question about the Monty Hall problem and Monty's knowledge of the probability distribution several months ago. I got some good answers and this one in particular helped me gain some ...
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Deriving the log-likelihood function for ACD (Autoregressive Conditional Duration) models

I am trying to understand the procedure that is shown in this survey to obtain the log-likelihood function to estimate the parameters for an ACD model: PACURAR, Maria. Autoregressive conditional ...
Residual Claimant 's user avatar
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Gambler ruin's: Probability of k consecutive win before j consecutive loss

Assume that a stock has a probability of $p$ to win, a probability of $q$ to lose, and a probability of $(1-p-q)$ to remain every day. What is the probability of $k$ consecutive wins occur before $j$ ...
Zhihao Xu's user avatar
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Transformer model conditional probability distribution of sub-sentences

I have a simple transformer model (decoder only) which is trained on some dataset containing sentences to do next-word prediction. The model captures a probability distribution $P_{\theta}(\mathbf{a})$...
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How to Compute the Posterior Distribution of Covariance matrix in a Matrix Normal Model with Inverse Wishart Prior

I am working on a time series model involving Kalman filters and smoothing to estimate state variables $Y_i$. The part of model is structured as follows: $Y_1, \ldots, Y_n$ are iid. $Y_i \sim \mathcal{...
Ayden Frost's user avatar
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Proving $E[Y],E[X] = \infty$ given their conditionals are Poisson distributed

Let: $$X \mid Y = y \sim \text{Poi}(a+y)$$ $$Y \mid X = x \sim \text{Poi}(a+x)$$ for some $a >0$. Are $E[X],E[Y] = \infty$ ? I tried: $$E[X] = E[E[X\mid Y]]$$ which , with the symmetry of the ...
lohe's user avatar
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Marginalisation with respect to arbitrary distribution

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space. Let $X: \Omega\rightarrow\mathcal{X}$ and $Y: \Omega\rightarrow\mathcal{Y}$ be random variables. Now, I know that $$\int_{C}\mathbb{P}_{...
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Data generation processes that "vary slowly enough"

I am considering some abstract data generation processes producing data by first drawing features $x_{1:n}\sim p(x)$ iid, and then drawing the responses $y_i\sim p(y\mid x_i)$ iid, where $p(y\mid x)$ ...
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Doubt regarding limiting distribution on Vasicek model

I was reading an article from Vasicek where he's concerned about deriving the limiting loss probability distribution on a credit risk model with 2 factors. I am here presenting a somewhat different ...
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Distribution of the random variable $\mathbb{P}(Y|X)$

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and let $X:(\Omega, \mathcal{A})\rightarrow(\mathcal{X}, \mathcal{F})$ and $Y:(\Omega, \mathcal{A})\rightarrow(\mathcal{Y}, \mathcal{G})$ ...
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Exercise similar to the ballot box model

Consider the following problem A box contains 10 bulbs of which 6 are new and 4 are used, 4 bulbs are selected at random without replacement until a used one is removed. I want to determine the ...
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Modelling the joint pmf of 2 correlated variables as p(x)*pmf(E(y|x))

Let x,y be 2 correlated counts. We want to model the joint pmf p(x,y). We know that p(x,y) = p(x)p(y|x) = p(y)(x|y). However, what happens when we don't know y|x, but we can estimate E(y|x)? Can't we ...
Dead Alive's user avatar
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In general, does $P(R|N)>P(R)$ implies $P(N|R)>P(N)$ ? Seems to be confirmed by Venn diagrams example but contradiction with Bayes' theorem

Let $R$ and $N$ be two dependent events, such that $P(R|N)>P(R)$. Does this implies that $P(N|R)>P(N)$? The odds are this is true (and equivalently $P(R|N)<P(R)$ implies $P(N|R)<P(N)$). I ...
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3 answers
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Law of the unconscious statistician for conditional expectation and pushforward measure of conditional distribution

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X:(\Omega,\mathcal{A})\rightarrow(\mathcal{X}, \mathcal{F})$ and $Z:(\Omega, \mathcal{A}) \rightarrow (\mathcal{Z}, \mathcal{G})$ ...
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Explicit sampling with a random conditional covariance matrix

Let $\mathbf X = (x_1,\ldots,x_N)$ be a zero-mean random vector which has conditional covariance $$ \Sigma_\xi \equiv \operatorname{cov}[x_i,x_j| \xi]=\frac{1}{N}\left( \sigma^2 \delta_{ij} + \sum_{\...
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Sufficient conditions for a conditional probability statement

Consider the random variables $Y, X, D, \epsilon_1, \epsilon_0$, where $X$ has support $\mathcal{X}$, $D$ is binary, $(\epsilon_0, \epsilon_1)$ are independent of $X$ and with full support on $\...
Star's user avatar
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How to derive conditional distribution function on range from conditional distribution function on a single value

Suppose we know $$f_{X|Y}(x|y)$$ which is the conditional distribution function of $$X=x | Y=y$$ how to derive the conditional probability function of $$X=x|y_1 < Y < y_2$$? Should it be $$\int_{...
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2 votes
3 answers
158 views

Conditional probability, who's right?

i need help with a math problem. Me, and at least two teachers have all done it and gotten different results, so now I'm asking the wise people of the internet for help settling the debate. Here's the ...
Azrael's user avatar
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3 votes
1 answer
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The Math Behind the Conditional Probability of a Probabilistic PCA

I am trying to understand how to calculate the conditional distribution of probabilistic principal component analysis. This is explained in the book "Pattern Recognition and Machine Learning"...
CAM_etal's user avatar
2 votes
1 answer
33 views

conditional probabilities in chess

Let's say I am analyzing certain chess positions of a chess player (Magnus Carlsen, Lichess games when playing as white). We know that the 'unconditional probability' (empirically speaking) of this ...
Oscar Flores's user avatar
2 votes
2 answers
138 views

Distribution of a random variable conditional on its being a maximum or not

Consider the random variables $\epsilon_1,\dots, \epsilon_D$ defined on the probability space $(\Omega, \mathcal{F}, P)$. Assume they are continuous. Let $$ Y=\sum_{d=1}^D d\times \mathbb{1}\{\...
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Conditional probability exercise: is this the right way?

I have a Pandas DataFrame and I have to calculate a particular probability. I don't have problems with the code, but with the mathematical concepts I have to use. The DataFrame contains information ...
cicciodevoto's user avatar
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1 answer
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About the use of Bayes' rule for continuous valued random variables

I am currently studying the book "An introduction to statistical learning with application in Python" and I am currently at the part 4 of chapter 4 where they explain the general framework ...
Vincent's user avatar
1 vote
0 answers
42 views

How to handle "multiple" conditional probability like P(A|B|C)? [closed]

We know that P(A|B)=P(A,B)/P(B) but what if we want to find the analog for this "multiple" conditioning P(A|B|C)?
Alessandro Ghilardi's user avatar
14 votes
2 answers
250 views

How to obtain $p(x)$ given samples from $p(y|x)$ and $p(y)$?

Here, assume both $p(y\mid x)$ and $p(y)$ are too complicated to get closed forms, and we can only draw samples from them. Is there any way to estimate or draw samples from $p(x)$?
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Looking for terminology to describe a certain partial independence condition on conditional probability

I find myself in a position where for events $X,Y$ and $Z$, I might have $$ P(X|Y,Z) = P(X|Y)P(X|Z) $$ I don't know what to call this, and it's difficult to search for potential phrases, since all ...
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Parameterization of the reverse process in discrete state space paper

In section 3.3. of the paper Structured Denoising Diffusion Models in Discrete State-Spaces the author defines the following parameterization used in both the reverse and sampling process. $$p_\theta(...
stacker87's user avatar
1 vote
1 answer
107 views

Formal definition of sufficient statistic

Let $(\Omega_X,\mathcal{F}_X)$ and $(\Omega _T,\mathcal{F}_T)$ be measurable spaces. Let $\mathfrak{M}$ be a family of probability measures on $(\Omega_X,\mathcal{F}_X)$. Let $X:\Omega\to \Omega _X$ ...
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Determining Distribution for Conditional Probability

I have that the conditional probability density of $Y|X$ is as such $f_{Y | X} \propto x^{y - 1}(1-x)^{n-y-1}\alpha^{n-y}\beta^{y}$ where $\alpha, \beta$ are constants in $(0, 1)$, $x$ is a random ...
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1 answer
31 views

Conditioning once or twice?

Let's say we have two random variables $Z \in \mathcal{Z}$ and $X \in \mathcal{X}$ with joint density $p_{Z,X}(z,x)$ with respect to a base measure. The density is assumed to factor as $$ p_{Z,X}(z,x) ...
PAM's user avatar
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2 votes
1 answer
118 views

Borel-Cantelli lemma on conditional probabilities

In a probability space $\big( \Omega, \mathcal{F}, P \big)$, suppose $\{E_n\}_{n\in \mathbb{N}} \subseteq \mathcal{F}$ is a sequence of mutually independent events. By Borel-Cantelli Lemma, the ...
Sanae Kochiya's user avatar
1 vote
0 answers
16 views

Do Bayesian networks have any rules with regards to zero probability RVs?

I am currently learning about Bayesian networks through Berkeley's AI course. In a Bayesian network, each node encodes the conditional probability of the random variable (RV) represented by the node ...
user985091's user avatar
1 vote
0 answers
28 views

How to derive conditional posterior predictive distribution from definition of posterior predictive distribution in bayesian regression?

In my situation, I have a set of data points: $$ z_{0:n} = \\{ (x_0, y_0),\dots ,(x_{n-1}, y_{n-1}) \\} $$ I am trying to figure out how to derive the fully expanded form for the conditional posterior ...
QMath's user avatar
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5 votes
4 answers
185 views

Distribution of $Z^2 \cdot I(Z > 0)$ where $Z \sim \text{N}(0,1)$

When using the Likelihood Ratio test for testing particular hypotheses and attempting to obtain an size-$\alpha$ test, I run into the expression $$ \mathbb{P}\left( Z^2 \cdot I(Z > 0) > c \right)...
YessuhYessuhYessuh's user avatar
1 vote
1 answer
46 views

Calculating confidence interval in linear mixed effect model

Consider a simple linear mixed effect model (LMM) $$Y_{ij}=b_{i}+X_{ij}\beta+\epsilon_{ij},$$ where $b_i\sim N(0,\sigma_b^2)$, $\epsilon_{ij}\sim N(0,\sigma_e^2)$. Typically, one can estimate $\...
Fangzhi Luo's user avatar
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1 answer
28 views

Calculating Porbabilites in a Bayesian Network

I have got the following Network and porbabilites And I need to Calculate the probability $P(C=False)$ I triend using the formula for joint porability distributions $P(X_1,…,X_n)= \prod_{i=1}^{n}P(...
Josef Sigron's user avatar
1 vote
1 answer
39 views

If event $A$ is a union of elements of $S(A)$, then $\min_{Z\in S(A):Z\subset A}P(B\mid Z)\leq P(B\mid A)$ for any event $B$

Let $A$ and $B$ be events in a probability space and $S(A)$ a collection of events such that $A$ is a union of some elements in $S(A)$. How could I then conclude that the conditional probabilities ...
Cartesian Bear's user avatar
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25 views

The concept and notation about MLE(Likelihood) and MAP

In generally, we say that X1, X2, ..., Xi are from a certain distribution, which can be represented by f(x;θ), where θ is an unknown parameter. When I read content related to MLE or the Likelihood ...
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4 votes
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37 views

What's the conditional distribution of X given X = Y

I'm in a setting in which X and Y are both $Beta(0.5, 0.5)$ and indipendent, that is $$f(x, y) = \frac{1}{\pi^2\sqrt{x(1-x)y(1-y)}}$$ What is the conditional distribution of X given Y = X? Normally i ...
Ivan's user avatar
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28 votes
4 answers
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Coin flip game: HH vs HT in a sequence of flips

An interesting thought experiment involving flipping a fair is going around X/Twitter: Flip a fair coin 100 times—it gives a sequence of heads (H) and tails (T). For each HH in the sequence of flips, ...
David B's user avatar
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1 vote
1 answer
37 views

conditional expectation of univariate normal given realization of multivariate normal

Consider the random variable $\textbf{x} = (x_1, x_2, ..., x_N)$ where $x_i \sim N(\mu_i, \sigma_i)$ for $i=1,2,...,N$ and $\textbf{x} \sim N(\mu, \sigma)$ where $N$ stands for normal distribution, $\...
user9875321__'s user avatar
6 votes
1 answer
498 views

Does independence almost everywhere imply independence?

Let be $X$ and $Y$ two random variables such that, for any event $A$, $P( X \in A \mid Y) = P(X\in A)$ with probability 1. Can I conclude that $X$ and $Y$ are independent ?
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What is $P(B)$ given $P(D), P(C), P(C|D), P(B|C), P(B|D), P(B|CD), P(B|C^cD^c)$ [closed]

Given the known values, $$P(CD) = P(C|D)P(D)$$$$P(C^cD) = P(C^c|D)P(D) = (1 - P(C|D))P(D)$$ Then using these values we find, $$P(D|C) = \frac{P(CD)}{P(C)}, \, P(D|C^c) = \frac{P(C^cD)}{P(C^c)}$$ This ...
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How to calculate a conditional value given two normally distributed variables and their correlation?

Assume we have two variables, Height and Weight. Each is normally distributed, and they are correlated. Assume the correlation was 0.75 for a specific example. Given a particular Height, can we ...
David vun Kannon's user avatar
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32 views

True or False: If the distribution of Y|X is normal, then the regression of Y on X must be both linear and homoscedastic [duplicate]

I'm trying to interpret an early and pretty dense (to me) paper on the theory of linear regression: Bartlett, M. S. (1934). On the theory of statistical regression. Proceedings of the Royal Society of ...
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