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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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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 [closed]

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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1answer
18 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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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
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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 ...
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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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1answer
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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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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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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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23 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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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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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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1answer
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Setting boundaries for calculating $P(Y/X>2)$ choosing $dx/dy$ order [duplicate]

Given two independent variables $X$ and $Y$, with marginal pdfs $f_X(x)=2x, 0 \le x \le 1$ and $f_Y(y)=1, 0 \le y \le 1$, calculate $P(\frac{Y}{X} > 2)$. So this can be written as $P(Y>2X)$, ...
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49 views

Why do we take a function of x as the limit of integration over y while calculating the marginal pdf of x?

I searched through similar questions but couldn't find one answering my question. I know the following is the way of finding marginal pdf from joint pdf. $$ f_x(x)= \int_{-\infty}^{\infty} f_{x,y}(x,y)...
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1answer
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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 ...
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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 ...
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1answer
64 views

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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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1answer
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$k$-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_{(...
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1answer
75 views

joint PMF in a class of n students

A class of n students takes a test in which each student gets an A with probability p, a B with probability q, and a grade below B with probability 1 − p − q, independently of any other student. If X ...
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1answer
66 views

PMF and independence with two discrete random variables?

Each of n people (whom we label 1, 2, . . . , n) are randomly and independently assigned a number from the set {1, 2, 3, . . . , 365} according to the uniform distribution. We will call this number ...
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Probability of successful match with two faces in image

A photo containing faces of two different people is compared to labeled images of faces in database. The probability of a match on the first person is .70. The probability of a match on the second ...
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Joint Probability Distribution and covariance

If $$f(x,y)=1/4 $$ $x=-3,y=-5; x=-1,y=-1; x=1,y=1; x=3,y=5. $Find cov (x,y). I know the formula for cov (X,Y) but I'm stuck at finding E (x) and E (y).
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32 views

Marginal Distribution from Bivariate Distribution Matrix

I am doing some practice problems to prepare for my statistics exam, and I just want to know if my reasoning is correct on one problem, and if not, I want to know how I should reason through this. The ...
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1answer
83 views

How to compute a 2D distribution function from its density?

Suppose that we have two random variables $X, Y$ with a joint probability density function $$f(x,y)=1,\ -y\lt x\lt y,\, 0\lt y\lt 1.$$ How can I calculate the cumulative joint probability function $...
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2answers
52 views

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....
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1answer
31 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
74 views

Joint cumulative distribution of independent random variables

X,Y,Z are non negative random variables which are independent and uniformly distributed in [0,1] and let $\alpha$ be a given number in [0.1]. Now how to compute $\text{Pr}(X+Y+Z>\alpha \;\;\; \&...
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Joint distribution of a two part model

Let $ Y $ be a random variable defined on $ (0, +\infty) $. In a univariate two part model, the distribution of $ Y $ is defined as follows \begin{equation*} g ( y_i ) = \left\{ \begin{array} { ...
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I am confused between selection of limits for joint distribution [duplicate]

I am given the following joint distribution of two random variables X and Y: $$ f(x,y)=2y \\ 0<y<1, 0<x<y $$ Now I need to find the marginal PDF of X but how do I ...
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What is the Joint Density Function of a Three-Level Mixed-Effects Model?

This is a follow-up question to a question I posted earlier. Obviously, maximum-likelihood estimation of mixed-effects models requires the joint density function. Let us assume a two-level mixed ...
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1answer
49 views

Multi-dimensional CDF on a discrete support

Suppose I have two discrete-support random variables, $X$ and $Y$. They have joint CDF $F(X,Y)$. If I want to find $\Pr(a \leq X \leq b , c \leq Y \leq d)$. It is obviously not: $F(b ,d)-F(a-1 ,...
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2answers
128 views

Deriving the joint probability density function from a given marginal density function and conditional density function

We have a multivariate normal vector $\mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$, where $\mathbf{X} = \left[ \begin{matrix} X_1 \\ X_2 \\ \dot\\\\ \dot\\\\ X_n \end{matrix} \...
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GMM model of the joint distribution from multivariate marginals

I have two multivariate Gaussian variables $X \sim \mathcal{N}(\boldsymbol {\mu}_X \in \mathcal R^d,\boldsymbol {\Sigma}_X \in \mathcal R^{d \times d})$ and $Y \sim \mathcal{N}(\boldsymbol {\mu}_Y \...
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2answers
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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, ...