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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Rewrite joint probability as product of marginals when all the probabilities are $1$ or $0$

I have a question about the possibility of rewriting a joint probability as the product of the marginals when all the probabilities can only take value $1$ or $0$. I start with introducing some ...
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102 views

Gibbs algorithm using negative binomial produces NAs

I have the following full conditionals distributions: $$ X_2|X_1=x_1\sim Bin(x_1,p)\\ X_1|X_2=x_2\sim NegBin(x_2,p) $$ So I'm using the following code to generate a sample from each one: ...
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Conditional distribution of a function of a random vector given conditional distribution of random vector

Let $\mathbf{X}=(X_1,...,X_n)^T$ be a multivariate normal distribution. Now we have $\mathbf{Y}=(Y_1,...,Y_n)^T$ defined by $Y_i = e^{X_i}$. Let $\mathbf{Y^1}, \mathbf{Y^2}$ be partitions of $\mathbf{...
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172 views

Permutation testing to check statistical significance of 2D distribution

I have a bunch of 2D distribution/histograms that represent variable X against variable Y, across several conditions. The histograms look something like: For each condition (each data summary as ...
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Intuitive explanation of “density generators”?

I was reading through Meucci's Risk and Asset Allocation (2005), when I happened upon the concept of a "density generator", which I have not been able to find good explanations for anywhere online, ...
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The joint distribution of two different sums of the same independent uniform random variables

Given $k$ independent, uniform random variables $X_i \sim U(-a,a)$, $i=0,\dots k$, and two sets of coefficients $\{\alpha_i\}$ and $\{ \beta_i\}$, let $U = \sum_{i=1}^k \alpha_i X_i$ and $V = \sum_{i=...
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Conditional probability density function (PDF) of bivariate normal distribution

Let $X$ and $Y$ have bivariate normal PDF with correlation coefficient $\rho$, i.e.,: $f(x,y)=\frac{1}{2\pi\sigma_X\sigma_Y\sqrt{1-\rho^2}}\exp{(-\frac{1}{2(1-\rho^2)}[\frac{(x-\mu_X)^2}{\sigma_X^2}+\...
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1answer
24 views

Computing a marginal distribution of a joint involving a delta function

Suppose that we have four continuous random variables $x,y,z,$ and $v$ and we want to compute the following integral: $$\int f(x\mid y)f(z\mid x,y)f(v\mid z,x,y)\,dx$$ There are a few conditions: $...
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93 views

Distribution of the Sum of 2 Rayleigh Distributed Values

Problem I have been trying to determine the distribution of the sum of 2 values sourced from a Rayleigh distribution with the parameter $\sigma$. Unfortunately, I am unable to match a computer ...
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393 views

Conditional Probability vs Conditional Probability Distribution

I am having hard time interpreting the relationship, if any, between conditional probabilities vs. conditional probability distributions, in particular, regarding the number of random variables ...
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MLE of joint normal covariance matrix under the constraint that the covariance is rank deficient

Given $n$ samples of a joint normal distribution of dimension $d$ with zero mean, $n>d$. I would like to ML estimate the covariance matrix, under the condition that the covariance is of rank $r<...
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How to appropriately apply activity probability distribution to location probability distribution?

Given the probability that an individual was in a particular area during an hour of the day and the probability that the sample population was doing an activity at an hour of the day, how would one ...
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22 views

Multivariate T distribution?

Suppose $(Y_1,Y_2)^T \sim N((0,0)^T,\Sigma)$, where $$ \Sigma = \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{pmatrix}. $$ $T_1= \frac{\bar{Y_1}}...
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Conditional probability of n sequential time series points

I have hourly road traffic volume data. While I assume the volume data can be defined as normally distributed, there is some correlation. i.e. The traffic in the past hour may affect the traffic in ...
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48 views

Learn this distribution from samples? What is the sample complexity?

$\newcommand{\norm}[1]{\left\lVert#1\right\rVert}$ We have an $n$-variate distribution $X\in\{0,1\}^n | \sum_i^n X_i = k$. Or, in other words, we are guaranteed that only $k$ variables will be $1$ in ...
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Have $(U,V)$ be a pair of Bivariate Gaussian variables with mean $0$, variance $1$ and $Cov(U,V) = p$ where 0 < ρ < 1 [duplicate]

Have $(U,V)$ be a pair of Bivariate Gaussian variables with mean $0$, variance $1$ and $Cov(U,V) = ρ$ where $0 < ρ < 1$ I'd like help finding the density of $U+V$ So far I have tried to use $...
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79 views

Meaning of the expression $P(\theta_1,\theta_2,\theta_3) \propto 1$?

I am reading this statistics books and it keeps on repeating same expression over and over again $P(\theta_1,\theta_2,\theta_3) \propto 1$ What does the above expression mean?
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374 views

Making inferences from Bayes networks and CPTs

So I'm practicing working with Bayes Networks and conditional probability tables and I feel like some of my numbers simply don't make sense. Here's the situation: I have a bag of three different ...
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28 views

Joint distribution of maximum and minimum of a bivariate normal distribution

Suppose $X = (X_1,X_2)^T \sim N(\mu, \Sigma)$, where $\mu =(\mu_1,\mu_2)^T$ and $ \Sigma= \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{pmatrix} $...
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What are the values of output $g_i$ in Bengio's paper “Taking on the Curse of Dimensionality in Joint Distributions Using Neural Networks”

Figure 2 of Bengio's paper "Taking on the Curse of Dimensionality in Joint Distributions Using Neural Networks" describes a neural network structure for estimating a joint probability distribution. ...
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1answer
837 views

Probability that one random variable is larger than another with known correlation

Let's say I have a normally distributed random variable $X_1$ with known standard deviation $\sigma_1$ and $E[X_1]$ is $0$. Let's say I have another variable with known standard deviation $\sigma_2$ ...
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1answer
35 views

Decomposing a random variable into marginals and copula

I’m having trouble getting understanding how to actual construct a copula, from my understanding it captures the purely joint features of a joint distribution. I’ve been working with the following ...
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Interpretation the law of total expectation

For a function of two RVs we have: $$ E[g(X,Y)] = E[\underbrace {E\{ g(X,Y)|Y\} }_{It\space is\space a \space function \space of\space Y}] = E[h(Y)] $$. For each $Y=y$, the inner expectation is the ...
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What is Supremal/Infimal Convolution

I'm currently reading a paper which mentions supremal and infimal convolutions. As I understood, they are upper and lower bounds for a joint distribution. One of the formulas in that paper is as ...
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187 views

How to find joint confidence interval for a bunch of normal distributed samples

Suppose there are two samples A and B. A has average $u_{1}$ and standard deviation $s_{1}$, B has average $u_{2}$ and standard deviation $s_{2}$. We know they come from two independent normally ...
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Distribution of Conditionally Dependent Random Variables

How would I find the distribution (or approximate distribution) of $X_t + Y_t$ given $$X_t\sim \operatorname{Normal} (0,1)$$ $$Y_t\mid X_t\sim \operatorname{Normal} (f(X_t),1)$$ where $f(\cdot)$ is ...
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1answer
47 views

The joint pdf of sample maximum and sample mean for uniform distribution?

Assume $$\{X_i\}\stackrel{\mathrm{i.i.d.}}{\sim} \mathcal{Uniform}(0,1)$$ Find the joint p.d.f. of $$X_{(n)} \hat= \max \{X_1,X_2,\ldots,X_5\}\quad\text{ and }\quad \bar X\hat=\sum^n_{i=1}{X_i}$$ ...
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692 views

Upper bounds for the copula density?

The Fréchet–Hoeffding upper bound applies to the copula distribution function and it is given by $$C(u_1,...,u_d)\leq \min\{u_1,..,u_d\}.$$ Is there a similar (in the sense that it depends on the ...
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19 views

Change of variable technique for conditional distribution

This might be a naive question, but could someone please tell me how the change-of-variables technique applies to conditional cases? My intuition tells me that there is no difference. The change-of-...
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Expected function evaluation of random variable w.r.t. different distribution

Suppose I have two continuous random variables on the same domain, $\xi \sim \mathbb{P}, \xi' \sim \mathbb{Q}, \in \Xi$ and joint probability $(\xi, \xi') \sim \Pi \in \Xi^2$ . Now I would like to ...
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Interpreting joint density distribution and contours

I have a landslide model and generated 1000 bootstrap samples of coefficients for 2 predictor variables (slope and log10_carea) using glm. I have created a ...
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3k views

Expected value of a marginal distribution when the joint distribution is given

I am asked to find the expected value of a vector of two random variables when the joint density is given. Is the recipe for solving this problem: Find the marginal distributions Find the expected ...
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1answer
19 views

Interesting variant on discrete probabilistic problem

Suppose $X \sim U(0,1)$ and $Y$ and $Z$ are random variables that depend on $X$. I've solved a problem where $Y$ and $Z$ are discrete (binary) and so finding the joint pmf just amounts to calculating ...
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33 views

How to evaluate double Integral with importance sampling

I am trying to recreate the Bayesian Hierarchical Clustering algorithm using Python. The example in section two requires evaluating the following double integral (univariate case): \begin{align} p(...
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The joint distribution of Y=AX and Z=BX given a projection matrix A and residual maker matrix B, and a random vector X with known pdf?

This question follows on from a previous question I asked which was answered. It turns out my question lacked some important details, which was revealed by the answer posted on that thread. This is ...
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1answer
39 views

How to derive the joint distribution of Y=AX and Z=BX given a random vector X with known pdf?

Given a random vector $X \in \mathbb{R}^k$, with a known pdf given by $f_X$. If $Y, Z \in \mathbb{R}^k$ are defined by $Y = AX$, $Z = BX$, where $A,B \in \mathbb{R}^{k\times k}$ are different, given, ...
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create joint prob distribution or empirical relation for two variables

There are two variables, X1 and X2. The experimental study shows that they are highly correlated. Are there any reliable ways to create an empirical mapping(or equation) between X1 and X2. Assuming ...
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identifying which of $d$ normal distribution generated a given sample

I have $d$ Normal Distributions, $N_1(\mu_1, \sigma_1^2) \cdots N_d(\mu_d, \sigma_d^2)$. We pick one of the $d$ distributions with each distribution having a probability of $\frac{1}{d}$ of being ...
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522 views

Why do copulas need the i.i.d assumption for marginal distribution?

Does anyone know if are there some assumptions for Copula method? I heard from someone that the data should be i.i.d (independent and identically distributed). Let's say, if I want to capture the ...
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Line of Best Fit

If we have a dataset with two variables, X & Y, we can find the line of best fit using the empirical data (and whatever method suits you best). However, what if know the true joint distribution ...
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77 views

Find mgf from joint pmf

The joint pmf of random variables $ X$ and $ Y$ is given by $$p_{XY}(x,y)= \begin{align} & \frac{e^{-2}}{x! (y-x)!}\quad\text{if}\,\,\,x= 0,1,...y,\ y=0,1,... \\ \end{align} $$ Find its mgf. \...
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Joint Probability check

There are 6 players and 18 cards. Each of the 18 cards is numbered 1-18 (each is unique). Each player is dealt 3 cards. Players A and B are the first two players in the deal. The deal was a uniform ...
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1answer
32 views

Visualizing separability / independence

I’d like to visually ‘see’ the independence of random variables. I tried plotting f(x), f(y), and f(x, y) for independent and dependent pairs of variables. However, the difference is still not ...
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1answer
27 views

Conditional independence and joint distributions in graphical models

I'm reading Deep Learning by Ian Goodfellow and Yoshua Bengio and Aaron Courville. In chapter 3 about graphical models, to reduce the model complexity, we assume that certain conditional independence ...
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1answer
27 views

conditional pdf and joint pdf

I am looking at a description of a process that says $f(y|a_1,z,a_0) = \dfrac{f(y,a_1,z,a_0)}{p(a_1|z,a_0)p(z|a_0)p(a_0)}$ I am not sure if I follow this joint pdf, conditical pdf , p(.) relation. ...
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1answer
41 views

How do I obtain joint distribution of uniform random variables conditioned on a sum constrain?

Let us say, we have two random variables, x1 --> U(10,20) i.e. x1 is uniformly distributed between 10 and 20, and x2 --> U(20,40) i.e. x2 is uniformly distributed between 20 and 40. Moreover, it has ...
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1answer
46 views

How to find CDF of a function of continuous joint distribution from PDF of joint distribution?

Here's what I think I should proceed: Make a level curve for the function keeping the constraints given in the PDF of joint distribution in mind. Find the area of interest keeping the constraints ...
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1answer
24 views

Which curve to select for finding the CDF of a function of a continuous joint distribution?

I came across a question which required to find the CDF of a function of a continuous joint distribution: $W=XY$. The following is the joint PDF: $$f_{X,Y}(x,y)=\begin{cases}\frac{xy}{4000}&,\, ...
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23 views

Level curves and functions of pair of Random Variables?

I came across the following question: I tried solving it, the following is my 1st attempt:(2nd method at the end) 𝑃[𝑊≤𝑤]=𝑃[𝑋𝑌≤𝑤]=𝑃[𝑌≤𝑤/𝑋] And then I simply double integrated keeping the ...
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Minimum CDF of random variables

I know that the joint cumulative function of two random variables X and Y is defined as: $F_{X,Y}(x,y)=P(X≤x,Y≤y)$. How can I find the CDF for $F_{X,Y}=\{x,x\}$. In other words is what will be $Pr\{...