Questions tagged [joint-distribution]

Joint probability distribution of several random variables gives the probability that all of them simultaneously lie in a particular region.

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Confidence interval for a normal RV that its std is the mean of a different normal RV

Question: Suppose $X\sim\mathcal{N}(\mu_X,a),Y\sim\mathcal{N}(\mu_Y,b\cdot\mu_X)$ where $a$ and $b$ are known, but $\mu_X$ and $\mu_Y$ are not. What confidence bounds can we give for $\mu_Y$ from one ...
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How to measuring the “correlation” of two sets of datapoints using copula’s

Say I have 2 sets of data points $(x_1,x_2,…,x_n)$, $(y_1,y_2,…,y_n)$ which are the log-returns of two different stocks, $S^X$ and $S^Y$, I have no reason to believe their log-returns are normally ...
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How MLEs obtained by fitting joint distribution fare against MLEs obtained by fitting conditional ones?

Let $(X,Y)$ be a bivariate r.v. with a known joint PDF $f(x,y \mid \theta)$ and conditional PDF $f(x \mid y, \theta)$, where $\theta$ is some parameter vector (let's assume here that $f(x \mid y, \...
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Hoeffding’s formula for Locally most powerful rank tests

Suppose we have a testing problem with $$H_0: X_1,X_2, . . . ,X_n \ \text{are i.i.d. random variables with a continuous cdf} \ F(x) \ \text{and pdf} \ f(x)$$ and $$H_1: X_1,X_2, . . . ,X_n \ \text{are ...
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On the asymptotic joint distribution of two weighted Bernoulli sums

Let $\theta_i$ be a Bernoulli random variable of success probability $p$. Given two fixed sets $A = \{a_1, a_2, ..., a_N\}$ and $B = \{b_1, b_2, ..., b_N\}$ where $a_i$ and $b_i$ are constants. For ...
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An impossible distribution

Some days ago another user posted a question which was something like this: $$ A \sim U(0,4)$$ $$B \sim U(0,6)$$ $$A - B \sim U(-4,4)$$ The question was originally to find the distribution of A ...
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Conditional expectation of Poisson, conditional on Poisson sums

Consider independent Poisson random variables $X_1\sim \text{Poisson}(\alpha_1)$, $X_2\sim \text{Poisson}(\alpha_2)$, $Y\sim \text{Poisson}(\lambda)$, and suppose $Z_1=X_1+Y$ and $Z_2=X_2+Y$. I want ...
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Join distribution of independent random variables that aren't conditionally independent

I am asked to give an example for a joint distribution of three random variables, $U$, $V$ and $W$, where $U$ and $V$ are (unconditionally) independent but are NOT conditionally independent given $W$. ...
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Probability of a certain final score in soccer when you know each of the two teams' probabilities to score a certain number of goals

Let's suppose that you have data from about $2000$ soccer matches from a certain league. Your data shows that the home team's probability of scoring $0$ goals is $0.245263157894737$. The away team's ...
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How are the joint distribution and dependency related? [closed]

Here are some notes about copula functions, Copula is a probability model that represents a multivariate uniform distribution, which examines the association or dependence between many variables. Put ...
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Form of a joint distribution table of random vectors and missing vectors

I am trying to follow this lecture on variational autoencoders. When talking about random observed data $o$ with missing components $m$ (min 14:10) he states that to calculate the log-likelihood of ...
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Using Law of iterated expectations, I want to calculate mean of Y, E(Y)

I obtain insights into calculating the conditional mean and variance of Y given X, denoted as E(Y|X) and Var(Y|X) respectively. Building upon this knowledge, I want to answer the follow-up question ...
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$U$-statistics and their limiting distributions

Let $X_1,X_2, . . . ,X_n$ be i.i.d. observations from a continuous distribution $F$. Consider the parametric function $\mathbb{P}([\text{min}(X_1,X_2) > X3])$. Find the U-Statistics and its ...
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Converting an integral into a probability of some event

Suppose that $X_1, X_2, .....X_n$ are iid random variables from some continuous distribution $F$. Show that $$\int_0^{\infty}(1-F(s+t))f(s)ds=\mathbb{P}(X_1>X_2+t, X_2>0)$$ $$$$Consider the ...
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Distribution of $\max_i \bar{X}-X_i$

Let $X_1, \ldots, X_n$ be i.i.d. random variables from the standard normal distribution and let $\bar{X} = \frac{1}{n}\sum_{i=1}^n X_i$ be their sample mean. I'm interested in the distribution of the $...
Theo Mary's user avatar
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Which model should be used to analyze repeated measures in the context of survival meta-analysis?

I am in the following context: each patient has at least 2 measurements, but possibly more (up to 5 measurements), of a given biomarker. The first measurement is taken at baseline, but the date and ...
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Finding joint probability distribution of standard normal with constraints

Let $X, Y \mathop{\sim}\limits^{iid} N(0,1)$. a) Suppose $X < Y$, find the joint pdf of $X$ and $Y$. b) What is the joint pdf of $X$ and $Y$ if $X = Y$? We know that $f_X(x) = \frac{1}{\sqrt{2\pi}}...
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Analytically estimate joint Von Mises distribution parameters from multiple underlying distributions with arbitrary weights

Given a set of n one dimensional (circular) Von Mises distributions, it is possible to randomly sample each distribution (with a different weight, ...
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Sum of Independent Laplacian Variables

I have $N$ Independent Random variables Laplacian distributions with $\mu=0$ and positive $b=\sigma^2/2$. I also have dominant random variable $(X_s)$ with Laplacian distribution with $\mu=0$ and $b=\...
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Joint and conditional probability with Poisson and Binomial distributions

I'm having a hard time trying to figuring out how to resolve a problem that involves a Poisson distribution and a Binomial distribution. Let's suppose that the total number of offspring (sons and ...
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How come the Bayes Theorem formula results in different probabilities that are verifiable using manual counting?

I assume Baye's Theorem is expressed as either: $P(B|A) = \frac{P(A|B)*P(B)}{P(A)}$ or $P(A|B) = \frac{P(A) * P(B|A)}{((P(A) * P(B|A)) + (P(A') * P(B|A'))}$ The tutorial problem was: Assume ten ...
Joachim Rives's user avatar
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Transforming data with a fitted distribution function

I have a bivariate dataset on $[0,1]^2$ in which I am interested in fitting a joint distribution. I fit a Gaussian copula but am unsure how to judge if it's a good fit. I tried transforming my data ...
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How do I generate a sampling distribution of a joint logistic-normal distribution in R?

I am working on a problem related to travel distances of people. I will use leisure trips (this includes day trips or multi-day trips) as an analogy. Focusing purely on the one-way travel distance, ...
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When transforming t in the joint probability density f (x, t), i.e. t=1/(mt+1), how does the marginal probability density f (x) transform?

for example, if: $$\begin{aligned} f_{X_{pi},T_{pi}}(x,t|*)=& \begin{aligned}\frac{\pi s^2}{\alpha_{pi}^2}\exp\left(\frac{\alpha_{pi}(x-\frac12)\nu_{pi}}{s^2}-\frac{\nu_{pi}^2}{2s^2}(t-\tau_p)\...
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Joint probability based on assuming linear relationships

I am a statistical novice trying to work something out which is probably basic. Or impossible. I have discrete probability distributions for two random discrete variables $A$ and $B$. I want to find ...
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Dependence or independence of three random variables

Consider I have three random variables A, B, C. I know that A depends on (B,C). Can I always deduce that it implies that A depends on B and also A depends on C? I mean does it implies that neither A ...
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how to calculate conditional probablity when one even'ts occurance is dependent on mutiple events

I have X, Y, and Z all as binary variables, values either 0 or 1. Y and Z are and got values of P(Y = 1), P(Z = 1), P(X = 1|Y = 1, Z = 1) , P(X = 1|Y = 1, Z = 0) and P(X = 1|Y = 0). here I need to ...
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Expected Squared Error Derivation of Least Square Model (in The Elements of Statistical Learning Book)

I'm studying chapter 2.5 of The Elements of Statistical Learning by T.Hastie. Here, they assume the ground truth relation between Y and X as $$ Y=X^T\beta +\varepsilon, $$ where $\varepsilon\sim \...
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Analysis of multivariate ranking data

I have data on companies, each company ranks how important are the following 4 elements (price leadership, quality, innovation, and customization) for its competitive strategy. There are 4 dependent ...
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Learning over non-independent joint distributions

For integer $n\geq 1$, I have a "goodness" function $f_n(F)$ that takes as input a given joint CDF $F$ of $n$ variables, and spits out a number in $[0,1]$ on how "good" $F$ is. The ...
AspiringMat's user avatar
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Deriving covariance of joint distributions of MVN [Linear Gaussian systems?]

Let $z$ ∈ R^L be an unknown vector of values, and $y$ ∈ R^D be some noisy measurement of z. We assume these variables are related by the following joint distribution $p(z) \sim N(z|\mu_{z}, \Sigma_{z})...
Kevin JJ's user avatar
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Existence of Distribution with Given Multivariate Marginals

Consider discrete random variables $X_1,\cdots, X_n$, and let $D$ be their joint distribution. For each subset $S\subseteq[n]$ let $D_S$ be the marginal distribution $(X_i)_{i\in S}$. Fix $k<n$. ...
AAA's user avatar
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How to mathematically express the fact that the conditional probability $P(Y|X)$ can be independent of $P(X)$?

Mathematically, $P(Y|X) = \frac{P(X,Y)}{P(X)}$ and so $P(Y|X)$ must depend on $P(X)$. Since $P(Y|X)$ will change when $P(X)$ changes. However, consider this scenario: X = amount of red meat consumed ...
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Customer default rate - joint distribution, marginal distribution when sum not equal 1

hope this question fits here, any help would be greatly appreciated. Background: I´m working on a data generator for customer default data. The default rate shall depend on multiple categorical ...
z1rak's user avatar
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Proving Incompleteness of joint sufficient statistic

Let $X_1, ..., X_n$ be a sample from the continuous density $C~exp(-(x-\theta)^4)$ (for $ -\infty < x < \infty$) with $\theta$ as unknown parameter. Show that the minimal sufficient statistic is ...
Stats_Rock's user avatar
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How does Variational Autoencoder approximate the joint probability distribution?

I know that in Variational Inference the idea is to approximate the posterior P(z|x, y) and I know that Variational AutoEncoders (VAEs) use the idea of variational inference through neural network ...
Amir Jalilifard's user avatar
3 votes
1 answer
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Gaussian Markov Random Fields - Conditional distribution from jointly gaussian with given precision matrix

Suppose I have jointly normal random vectors $[\bf{v_1}, \ldots, \bf{v_K}]$' with mean $ \bf{M}$ and joint block tridiagonal precision matrix $ \bf{P}$: $$ \bf{M}= \begin{bmatrix}\bf{\mu_1} \\\ldots \\...
Giorgetto's user avatar
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Estimating joint probabilities across two datasets

I would like to estimate the joint probability of two variables from two different surveys conditional on other variables that the two surveys have in common. As an example I'm using data from this ...
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Find the conditional distribution from joint normal distribution with vec operators

I have two random matrices one on the top of the other: $ \begin{bmatrix}\boldsymbol{B_1} \\ \boldsymbol{B_2} \end{bmatrix}$. and they are both of dimension $k \times N$. I have that: $ vec\begin{...
Giorgetto's user avatar
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Upper bound for sum of dependent normal variables

I am having difficulties with the following problem: Assuming $X$ and $Y$ follow a bivariate normal distribution with $\mu = 0$ and $\Sigma=\begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix}$ ...
Coach's user avatar
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Probability distribution of the product of three dependent continuous random variables

Given the joint probability distribution for three dependent continuous random variables, I want to find a formula to compute the probability distribution for the product of these 3 random variables. ...
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Joint Distribution of Correlated Binomial and Negative Binomial Variables

Let $x$ and $y$ be correlated negative binomial distributions, $x\sim~NB(p_1,n_1)$ and $y\sim~NB(p_2,n_2)$ respectively with correlation $\rho$. Can we derive their joint distribution analytically? ...
anotherGUEST's user avatar
4 votes
2 answers
518 views

Finding the conditional expectation given the joint density function

Suppose a random vector $(X, Y )$ has joint probability density function $f(x, y)=3y$ on the triangle bounded by the lines $y = 0, y = 1 − x$, and $y =1+ x.$ Compute $E(Y \mid X ≤ 1/2 ).$ I'm ...
Tapi's user avatar
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Finding regression equations from joint distribution

Let $X$ and $Y$ be two random variables with joint probability density function $f(x, y) = 1$ if $− y < x < y, 0 <y< 1$ and $0$ elsewhere. Find the regression equation of $Y$ on $X$ and ...
Tapi's user avatar
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Question about joint cdf

we have that $P(X \leq x, Y \leq y) = \int \int_{s \leq x, t \leq y} f_{X,Y}(s,t) dsdt$ But how would for example $P(X \leq x, Y \geq y)$ Be defined? Would it just be: $P(X \leq x, Y \geq y) = \int \...
Parinn's user avatar
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1 answer
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$E[XY]-E[X^2]-E[Y^2]$, is there any special property?

Given probability distributions of random variable $X,Y$, without any additional assumptions, is there any nice representation or properties of the combination $E[XY]-E[X^2]-E[Y^2]$? If not, is there ...
user387393's user avatar
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Joint distribution of mistakes of phones

I am a bit confused with the following problem of probability: "We select 3 phones of a sample in which we know that the 60% of them contain one mistake and the 10% contain more than one mistake. ...
CharlesJA's user avatar
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3 votes
1 answer
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Is there a simple error in the answer key, or am I using the wrong approach to get $P(X<0.5)$

I am working on a problem that gives me a joint pdf: $$f_{x,y}(x,y) = 6xy, 0<x<1, 0<y<\sqrt{x} $$ I am asked to find $P(X < 0.5)$ with three decimal places. My approach was to integrate:...
gabiii's user avatar
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Probability that both the mean and sample variance are both covered by their respective confidence intervals?

I am given the question: "What is the probability that both the mean is in its confidence interval for confidence level a and the variance is in its confidence interval for confidence level a?&...
user386465's user avatar
1 vote
1 answer
267 views

When is an AR(1) process strictly stationary?

Suppose I have an AR(1) process $X_t=aX_{t-1}+e_t$, where $e_t$ is a white noise with zero mean and finite variance. Under what conditions do I have $\{X_t\}$ being strictly stationary in the sense ...
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