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).

Filter by
Sorted by
Tagged with
0
votes
0answers
18 views

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 ...
0
votes
0answers
18 views

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 ...
1
vote
1answer
44 views

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{...
0
votes
1answer
25 views

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)}-...
1
vote
2answers
34 views

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 = ...
1
vote
0answers
66 views

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 ...
0
votes
3answers
32 views

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 $P(y = -2 | x)$ ...
1
vote
1answer
58 views

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 ...
0
votes
2answers
27 views

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 ...
0
votes
0answers
13 views

Computing the bivariate distribution from trivariate?

I appreciate in advance for any suggestions. Patton. A (2008), Modelling asymmetric exchange rate dependence, Int. Econ. Review.
0
votes
1answer
32 views

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 ...
1
vote
1answer
16 views

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 ...
0
votes
0answers
8 views

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?
1
vote
1answer
33 views

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 ...
1
vote
1answer
12 views

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 ...
0
votes
0answers
12 views

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 $y_0$ ...
0
votes
0answers
52 views

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, .....
0
votes
2answers
77 views

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 ...
1
vote
1answer
38 views

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!
2
votes
2answers
82 views

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 $Y$ = ...
2
votes
1answer
54 views

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 ...
0
votes
0answers
16 views

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 ...
0
votes
1answer
23 views

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+...
0
votes
1answer
16 views

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 ...
0
votes
0answers
20 views

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 $X(t)$...
1
vote
0answers
42 views

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 (...
1
vote
0answers
14 views

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. ...
0
votes
1answer
32 views

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 |...
0
votes
1answer
37 views

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? -- What I've tried: I know that $$ P(R1&...
0
votes
1answer
50 views

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 $...
1
vote
1answer
266 views

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 ...
0
votes
0answers
15 views

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 ...
1
vote
1answer
17 views

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 ...
1
vote
1answer
35 views

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 $...
0
votes
1answer
37 views

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 ...
0
votes
0answers
17 views

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?
2
votes
1answer
37 views

How to calculate conditional expectation?

How can I calculate a conditional expectation like below?
0
votes
0answers
12 views

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 ...
2
votes
1answer
51 views

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 ...
2
votes
1answer
25 views

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-...
2
votes
2answers
26 views

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? $P(X|Y\cup Z) = P(X|Y) + P(X|Z)$
2
votes
1answer
28 views

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 ...
0
votes
0answers
11 views

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 ...
0
votes
0answers
26 views

How to estimate conditional CDF using python StatsModels

I am using statsmodels '0.10.1' to obtain conditional CDF of a 2 variable problem. ...
2
votes
1answer
32 views

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 $X$ ...
0
votes
1answer
59 views

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 ...
1
vote
2answers
49 views

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 ...
0
votes
1answer
22 views

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: ...
0
votes
1answer
40 views

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, ...