# Questions tagged [joint-distribution]

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

479 questions
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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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### probability distribution of two-dimensional samples [on hold]

Let's say, I have n samples that have the dimension of p-by-q (drawn from a continuous probability distribution). Here, p is the number of features $(X_1,...,X_p)$, and q is the sequential time vector ...
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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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### 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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### 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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### 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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### 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 ...
1answer
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### 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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### 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 ...
2answers
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### 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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### 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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### 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. ...
1answer
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### 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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### Finding the joint CDF using the joint PDF; why can't I do this?

Find the joint CDF of the independent random variables $X$ and $Y$, where $f_x(x)=x/2, 0\le x \le 2,$ and $f_Y(y)=2y, 0 \le y \le 1$. To do this, we can find the CDF separately for each of the ...
3answers
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### Compute $P(Y<3X)$ using joint PDF

I'm given a joint pdf $f_{X,Y}(x,y)=2e^{-x-y}, 0<x<y, 0<y$ and asked to compute $P(Y<3X)$. To do this, I let $Y=3X$ (the boundary) and found that the region of integration is under this ...
1answer
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### How to approximately sample an unknown non-parametric joint distribution given a complete set of partial conditional distributions?

This question is related somewhat to Bayesian networks. In a BN, you have a DAG (directed acyclic graph). By supplying the root nodes with a sample, you can then follow the directed arcs to sample ...
1answer
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### Finding the CDF given marginal PDF's; setting bounds

In this question, I'm having a hard time understanding how specifically to set the bounds for the CDF. Let $X$ and $Y$ be independent variables. Find the CDF of $W=Y/X$ using the marginal PDFs ...
0answers
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### Combining subjoint distributions to create a larger joint distribution

I am trying to construct large joint distributions through smaller joint distributions and I'm not sure how to approach the literature. I am curious if there exists a function which can take n ...
0answers
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### Optimize an objective based on a trained model

I want to find a joint optimal subset based on individual scores from a predictive model. Example: Say I have a set of customers and a set of products. And I have trained a model for predicting the ...
0answers
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### What's the probability that all three parts would fail within 2 years of each other? (joint PDF)

Suppose an instrument has three independent parts, all of whose lifetimes (in years) are modeled by an exponential pdf which is $f_Y(y)=e^{-y}, y>0.$ What's the probability that all three parts ...
1answer
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### Wavenet joint probability

As presented in the first article of Google Wavenet (https://arxiv.org/pdf/1609.03499.pdf) the model can approximate the joint probability of the whole sequence (raw audio waveform) using the chain ...
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### Having difficulty deciding limits of integration for a joint to marginal pdf

A joint pdf, $f_{X,Y}(x,y)=5$, is given with the following intervals: $-1<x<1$ $x^2<y<x^2+{1\over{10}}$ I am trying to find marginal pdf of $f_Y(y)$ but I am stuck. Trying for hours....
1answer
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### 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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### Identically distributed vs P(X > Y) = P(Y > X)

I've two related propositions which seem correct intuitively, but I struggle to prove them properly. Question 1 Prove or disprove: If $X$ and $Y$ are independent and have identical marginal ...
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### How to specify joint distribution when they are not jointly independent?

I have three variables, $X, Y, Z$ with marginal distributions $F_x, F_y, F_z$. I want to specify the joint distribution $F_{xyz}$ of $(X, Y, Z)$. I know that if $X, Y, Z$ are jointly independent, ...