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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Understanding three prisoners (Statistical Inference - Cassella and Berger)
There are quite a few questions regarding three prisoners riddle, but my question relating to a particular line of reasoning from Cassella and Berger, as highlighted in the image below.
Could you ...
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How to calculate Bayes factor for conditional probability?
I have a data set of 1000 drug-effect pairs. I am trying to identify which drug is most likely given the observed effect.
My original approach was to calculate $P\left(d_j | e_i\right)$ for each ...
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Distribution of sum of independent random variables using MGF
Assume you have $x_i \sim \operatorname{Bernoulli}(p_i)$ with $p_i \sim \operatorname{Beta}(\alpha,\beta)$.
and let $Z=X_1+ \dots +X_n$
and I wanted to show that $Z$, $Z \sim \operatorname{...
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Conditional Poisson Process
I cannot reach a correct answer and I don't know why. I am trying to calculate this by conditioning on $N(t)=n$ and I ended up with $e^{-At(z^s)}$. However, the correct answer is $\dfrac{e^{-At(z^s)}-...
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Does the joint pdf $f_{x, y} (x, y)$ equal to the conditional $p_{y | x} (y | x)$ for all random variables?
So I have this question where you are given two random variables, $X$ and $Y$. $X$ is a continuous random variable (represented as a mean) with a distribution of $Exp(1)$ (exponential with $\lambda = ...
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A simple question about expectations
@psboonstra This is a valid point. After posting it, I too realized that the question is ill-posed. I attempted to oversimplify a problem that I had encountered in finding a characteristic function of ...
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What methods can we use to predict probability distributions?
I'm wondering what methods we can use to predict a probability distribution. Essentially, given some observation $x$, I'm interested in calculating quantities such as $P(y = 3 | x)$ or $P(y = -2 | x)$ ...
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Consistent estimator of conditional expectation, when conditioning on binary variable
Suppose I have a sequence of i.i.d. random variables $\{Y_i,X_i,Z_i\}_{i=1}^n$ and $Z_i$ is binary.
Is the following a consistent estimator of $E(Y_i*X_i|Z_i=1)$ as $n\rightarrow \infty$? Under which ...
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Conditional expected value when conditioning on a set of values
Consider the random variables $Y,X,Z$ with supports $\mathcal{Y},\mathcal{X},\mathcal{Z}$, respectively.
Suppose that $E(Y|X=x,Z=z)=0$ for each $(x,z)\in \mathcal{X}\times \mathcal{Z}$.
Does this ...
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Computing the bivariate distribution from trivariate?
I appreciate in advance for any suggestions.
Patton. A (2008), Modelling asymmetric exchange rate dependence, Int. Econ. Review.
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Subdivide Z into $X_1,X_2$, s.t. $Z=X_1+X_2$ and ($X_1,X_2$) obey bivariate normal
For a simulation in a research project, I am trying to randomly "appropriate" (meaning subdivide into two components) known values of $Z$ into $X_1$ and $X_2$ such that $Z=X_1+X_2$ in a way that ...
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Notation for conditional density
Are $p(\mu \mid \sigma)$ and $p(\mu ; \sigma)$ equivalent?
I've seen the notation $p(b_i \mid T_i, \delta_i, y_i ; \theta)$ used to represent the posterior distribution for $b_i$. I am assuming that ...
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What's the latent random variable that explains Polya's urn?
According to De-Fenitte, exhangeable r.v's can be thought of as iid given a latent random variable. My question is: for a Polya's urn model, what would this hidden random variable be?
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Bayesian update vs optimization
Say I have a multivariate normal vector
$$ r \sim N(\mu , \Sigma ) \Rightarrow Pr \sim N(P\mu , P'\Sigma P ) $$
and I observe that
$$ Pr = Q $$
Now I can use Bayes rule to calculate the ...
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Deriving conditional independence from product rule of probability
I am reading Mathematics for Machine Learning and, in the Summary Statistics and Independence section, the author derives the conditional independence of two random variables given a third RV using ...
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Prove selection bias OLS
I have some questions regarding the following theoretical model and finding the selection bias. I have an idea but would like some guidance.
$y_0 = \alpha_0+X_i\alpha_1+\epsilon_0$
where $y_0$ ...
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Impute Missing Data Values with Mixture Models
Suppose I have a dataset with $o$ representing a collection of data dimensions with observed values and $d$ representing missing dimensions. The mixture model consists of discrete variables $Z = {1, .....
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Probability of a girl given 2 boys [closed]
I recently had a test on Probability Distributions and got this question wrong. Some help would be appreciated. If I recall correctly, the question was :
In a family of 4 children, the probability ...
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How do I factor this conditional probability?
I am having a brain freeze. Could you show the steps to get from line 1 to line 2?
Thanks!
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Given distribution of $X$ and $X|Y=y$, is it possible to find distribution of $Y$?
What the title says!
My intuition is NO since in Bayesian statistics we typically specify the prior and likelihood, and from those two we can compute the posterior and so on. We can interpret $Y$ = ...
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Estimate random effects for a new individual with a linear mixed effects model
Consider repeated observations $\mathcal{Y} = (y_{i,j})_{i,j}$ obtained for $p$ individuals ($1 \leq i \leq p$), at different time points $t_{i,j}$ $(1 \leq j \leq n_i$). The "random slope and ...
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Estimating mean of a normal distribution from a distribution of means
There are $i$ number of normal distributions $Y_i\sim N(\mu_i, 8^2)$. The means of all the $i$ distributions form another normal distribution $X\sim N(85, 2.5^2)$. Suppose a distribution $Y_i$ is ...
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Conditional probability of timeseries given history
How is the following interpreted?
Given two time series $X = \{x_1, \dots, x_n\}$ and $Y=\{y_1, y_2, \dots, y_n \}$, calculate probability of the next $x$, given the history $\mathbf y_d = \{ y_{n-d+...
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Interpreting the decision tree table
I am having hard time understanding how to interpret the table below.
It is clear that there are only 5 cases where the weather is rainy. But when I think of (Temp = Cool or Wind = Weak) I don't know ...
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On the transition probability distribution of Gaussian Brownian motion
I am having trouble understanding certain aspects of the following derivation. I'll first present it, and then follow up with questions. The derivation is as follows:
Consider a random variable $X(t)$...
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Conditional Probability Approximation
I have a complex problem that I can simplify into what seems like an application of Bayes theorem. The information I do have is something like this:
I know it rains 60% of the time in this location (...
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Derivation of Conditional Causal Probabilities
In Causal Inference in Statistics: an Overview, Pearl presents an equation describing distribution from a graphical model presented in figure 3:
The author arrives at equality (20) - see image above. ...
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Markov Chain Definition Question
This is a very basic question about Markov chains, but I am desperately lost. We are given the following definition of a Markov chain:
$P(X_{n+1}\in A | X_1 = x_1, ..., X_n=x_n) = P(X_{n+1} \in A |...
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Sum of two continuous random variables
Let R1 and R2 be two independent random variables, both with uniform density at the interval (0,2).
What is the probability of R1>1 given that R1 +R2<2?
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What I've tried:
I know that
$$
P(R1&...
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Conditioning on MGF
Suppose $Z_i$ is the total loss from all losses on policy $i$, where
$q_i=P(there\ are\ losses\ from\ policy\ i),\ i=1, \dots, n.$
Then $X_i$, the total loss on policy $i$ can be defined as $...
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What is the conditional expectation of the exponential functional?
Consider the function $g(W)=-e^{-W}$, where $W$ is some random variable s.t.$W=X+YZ$. Furthermore, it holds that all the random variables $X,Y,Z$ follow the normal distribution with the following ...
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How to implement single Imputation from conditional distribution?
In [*] page 264, a method of drawing a missing value from a conditional distribution P(X_mis|X_obs;Theta) which is defined as:
I did not find any code ...
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Rolling a dice probability given the history
Suppose that I rolled a dice once and got 3. I rolled it second time and got 4.
Is it true, that probability of getting 5 on third roll, given two previous attempts is not $\frac{1}{6}$? It should be ...
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Joint Prior Distribution for $Uniform(a,b)$ (a,b unknown) [closed]
I have some problems in figuring out how to address the following problem and some help would be welcomed.
Exercise
Consider a Continuous Uniform Distribution over the $
(a,b)$ interval where $...
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Why is this conditional dependency statement not correct?
I am taking an online Bayesian statistics course and here is a question from the quiz:
"Is the following statement correct? $$p(a∣b,c)=p(a∣b)p(a∣c)$$ when $b$ and $c$ are independent."
I thought it ...
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Gaussian Distribution [duplicate]
Assume we have two continuous Normal RV "X" and "Y".
how can I show the conditional PDF f(X|Y) and f(Y|X) is Normal?
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Do Conditional Independence Statements in Probabilistic Graphical Models Include Independence Statements?
Been reading Murphy and other books and trying to understand conditional independent statements of a graph: $I(G)$, or $CI(G)$ depending on the reference.
They mostly define a conditional ...
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1answer
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Simple Question On Independent Events
I'm reading DeGroot and it says:
A and B are independent if and only if Pr(A|B) = Pr(A) and Pr(B|A) =
Pr(B).
My question is:
Do both equations need to hold for the two events to be independent ...
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Are these two distributions independent?
In a probability cheatsheet, there's the following claim:
That $Z_1$ and $Z_2$ should be independent seems very unintuitive to me. It seems to me that if $Z_1$ is high, then $Z_2$ is low, and vice-...
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2answers
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What additional information would I need to calculate P(X|Y or Z) given P(X|Y) and P(X|Z)?
Is this always true when Y, Z are independent?
$P(X|Y\cup Z) = P(X|Y) + P(X|Z)$
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1answer
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Calculatiion of the joint probabilities
I was refreshing my knowledge regarding the definition of joint probabilities and read a page from the book: 'Econometrics for dummies' which gave the following example.
What I intended to do was to ...
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Deriving full conditional of ordered probit model (Bayes)
I have a question regarding the following exercise:
I am able to compute the complete (full) data likelihood function, the full conditionals of $y^{*}_i$ and $\beta$. However, I do not know how to ...
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How to estimate conditional CDF using python StatsModels
I am using statsmodels '0.10.1' to obtain conditional CDF of a 2 variable problem.
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Conditional independence problem for poisson random variables
I have this problem:
Let $X = V + W$ and $Y = V + Z$ where $V, W, Z$ are independent Pois($\lambda$) random variables.
I found that $Cov(X, Y) = Var(V) = \lambda$
It now asks to find whether $X$ ...
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Using the law of total expectation and the definition of the MGF to find the unconditional distribution
During my research, I encountered Jarle Tufto's answer in this question on the MGF of conditioned random variables:
The mgf of $Y$ conditional on $N=n$ is
$$ M_{Y|N=n}(t)=M_X(t)^n, $$
since ...
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Intutitive meaning behind the formal definition of sufficient statistic?
According to the definition of sufficiency, a statistic is sufficient for a parameter if the conditional distribution of $X$ given a value of statistic does not depend upon the parameter.
What I am ...
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Conditional probability table from deterministic relationships of two discetizied distributions - for Bayesian Networks
Consider a simple Bayesian Network of three variables A, B, and C. All of the variables are discrete variables between (0,1] that are discretized as below:
...
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Finding expected values from joint distribution
For my textbook, Introduction to Probability by Blitzstein and Hwang, I have the problem where I have the random variables $X$, $Y$, and $Z$ such that $X \sim N(0, 1)$. I am also told that, ...
