# Questions tagged [conditional]

This tag is ambiguous. Consider replacing it with a more specific tag such as [conditional-probability], [conditional-expectation], [conditional-random-field] or [conditional-independence].

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### Coherence of conditional probabilities

Dennis Lindley's paper The Philosophy of Statistics in 2001 includes the following 'simple' example of statistical coherence: "A set of uncertainty statements is said to be coherent if they ...
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### Deriving Bayes Rule from conditional probability [duplicate]

Bayes Rule and Conditional Probability look so similar to me. I'm having a hard time figuring out how to derive Bayes from the conditional probability equation. If I start with $$P(A,B) = P(A|B)P(B)$$,...
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### Conditional expectation, conditional on sum of weighted average of two iid RVs

I have an arbitrary distribution $F$, and two variables $z, x \sim F$. I only observe the weighted average $y = \alpha z + (1 - \alpha) x$. Conditional on $y$, what is the expected value of $z$? I ...
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### Possibility n independent random events ocuuring per year? An event is any piece of equipment failing per year

So I am an engineer. We have a method at our organization of defining the probability of an equipment failure based on frequency of occurrence. So for example we may categorize a piece of equipment ...
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### Conditional expectation versus correlation

Consider two random variables $X$ and $Z$. Suppose $E(X)=3$ and $E(X|Z=z)=0$ for some realisation $z$ of $Z$. Does this imply that $X$ and $Z$ are correlated? Does this imply that $X$ and $Z$ cannot ...
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### Is it possible that marginally independent random variables are conditionally dependent?

Suppose that $X,Y$ and $Z$ are random variables. If $X$ is independent of $Z$ and $Y$ is independent of $Z$, is it possible that $X$ is dependent on $Z$ given $Y$ and $Y$ is dependent on $Z$ given $X$?...
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### Prove P(A|B) = P(C|B)*P(A|C) + [1-P(C|B)]*P(A|C')

My textbook claims the following probability derivation, but I am still having trouble understanding where the formula came from. The derivation is as follows: ...
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### conditional covariance of $b$ and $\hat{\sigma}^{2}$

By definition of the normal linear model (i.e. standard linear regression under the assumption of gaussian noise with constant variance), we know that the regression coefficient vector $b$ is ...
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### How do I model conditional logit when some of the alternatives are not available to some individuals?

I am using conditional logit with discrete choices and random coefficients for some of the variables. There are 9 alternatives in my model that are the same for everyone in the sample. However, some ...
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### Reject the claim that a sports league's regular season performance can occur by chance

For a tournament where each of $K$ teams plays each other team exactly once, what is the probability that the most number of games any team wins is $N$? The answer to this question is to create ...
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### Goodness-of-fit for conditional logistic regression in a 1:1 matched case-control study

Dear Stackexchange community; I would appreciate if someone would guide me on this matter. On data analysis of a 1:1 matched case control study based on age and gender through using conditional ...
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### maximum likelihood with conditional distribution

Let $(y_i,x_i),(i=1,...,n)$ be observations with $y_i|x_i \sim iidN(0,\sigma^2(\theta,x_i))$. I want to estimate $\theta$ by maximum likelihood. But I cannot derive the likelihood function. If $x_i$ ...
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### Conditional expectation of two random variables

Suppose we have $X \sim Exp(4)$ and $Y \sim N(0,1)$ which are independent. What can we say about $\mathbb{E}[X|Y]$?
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### Logistic regression analysis or something else?

I have a dependent binary outcome and 6 independent variables. These are measured in the same group of people at two moments in time in a descriptive longitudinal study. I am assessing whether these ...
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### Poisson Process Conditional Probability computation

Given $N(t)$ a Poisson Process with generation rate $\lambda$ with $t_1<t_2$ and $N_2>N_1$ I'm looking for a way to express the following probability: $$P[N(t_2)>N_2|N(t_1)<N_1]$$ In ...
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### Interpretation of conditional Granger results

I have a question about the Granger package. How are the numbers obtained in the frequency and causal output of the conditional Grangers interpreted? Thank you.
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### Under which conditions does $X_1 \perp X_2, X_3$ imply $X_1 \perp X_2 | X_3$?

E.g. let us imagine we have $X_3 := X_1 \text{XOR} X_2$ with both $X_1, X_2$ being sampled from $\{0, 1\}$ with $p=0.5$. Then $X_1 \perp X_2, X_3$ but $X_1 \not \perp X_2 | X_3$. Are there conditions ...
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### Conditional expectation and variance of a poisson distribution, where $𝑋_𝑛$ is conditioned on $𝑋_{𝑛−1}$ [closed]

Suppose that $𝑋_0 \sim \text{Poi}(𝜇)$ (where $𝜇 > 0$ is fixed) and that for $𝑛 = 1, 2,…$ the distribution of $𝑋_𝑛$ conditional on $𝑋_{𝑛−1} = 𝑥$ is Poisson with parameter $𝑥$. Determine ...
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### is a pair of two conditional gaussian distribution imply a joint gaussian distribution

suppose x,y are two random variables, if p(x|y) is normal distribution, and p(y|x) is also normal distribution, can we say that p(x,y) is also joint normal distribution? i know that two normal ...
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### How to produce a nomogram for a conditional survival model?

Conditional survival is the survival probability after already surviving a predeﬁned time period. The formula used for conditional survival (CS) was: CS(x|y) = S(x + y)/S(x), where S(x) represents ...
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### is there a R function to do multiple comparison for all levels of a categorical variable by capscale (vegan) with a covariate? [closed]

First, I did DB-RDA for a community dataset with two factors and one covariate. The results showed the interaction effect was significant. Then, I need to do the pairwise comparison for all the levels....
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### Fisher Information for Cox Model

Actually, I'm working on a Statistical Genetics Article (Schaid and al,2010) in a retrospective likelihood context. In the article, authors present some result about conditional likelihood but I can't ...
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### Scenario Analysis with a GARCH model - conditional forecasts with hard restrictions on dependent variables?

Waggoner and Zha (1999), see reference below, developed an approach to produce conditional forecasts for VAR models with hard restrictions on the variables using Gibbs Sampling. As an example, they ...
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### Conditional expectations and variance - OLS

Apologies for starting a new post for such basic problems but I have an exam tomorrow and struggle to sort out two answers from last year's exam (2. and 3.). I gave as much context as possible: The ...
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### Conditional and density probability (normal distribution)

I am trying to solve the following problem: Suppose that $\mu\sim N(1,4)$ and $Y|\mu\sim N(\mu,1)$. Show that: \begin{bmatrix}Y \\ \mu \end{bmatrix} \sim N\bigg(\begin{bmatrix}1 \\ 1 \end{bmatrix},...
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### Inference problem, having issues translating problem from words into probabilistic terms

This is the question: There are particular readers for swipe cards, Under normal operating conditions, there is a small probability of 0.02 that swiping a card fails to open the entrance - some of ...
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### conditional distribution for normal distribution

Assume $X = (X_1, X_2 , ... ,X_n )$ is observed, where the $X_i$ are (independently) $N(\theta , 1)$. find the conditional distribution of $X_1,....X_{n-1}$ given $\bar{X}$. I am having a problem in ...
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### What is the posterior mean of $\mu$ given a randomly stopped i.i.d. observations from a Normal

Let's imagine I have a machine giving me an independent random number from a normal distribution $N(\mu,1)$ whenever I push a button. I have a stopping rule to decide how many samples to collect - I ...
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### Conditional Covariance Problem

Suppose we have independent (not necessarily identical) normally distributed random variables X, Y. If we're given that, upon sampling each variable, X is some multiple a of Y (i.e. x = ay), what is ...
### $k$-th order statistics when the value of $j$-th one is known
Suppose there are $n$ random variables $X_i,~i\in\{1,\cdots,n\}$ which are independently drawn according to a CDF $F$ and pdf $f$. Suppose also that we know one of the realization, say \$X_{(j)}=x_{(...