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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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Is this sentence referring to joint or conditional probability?

The following is a quote from my textbook, in a chapter discussing the Viterbi algorithm (Durbin, Richard, et al. Biological sequence analysis: probabilistic models of proteins and nucleic acids. 1st ...
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When $(X_1-X_0, X_1-X_2)\sim (X_2-X_0, X_2-X_1)\sim(X_0-X_1, X_0-X_2)$?

Consider a bivariate probability distribution $P: \mathbb{R}^2\rightarrow [0,1]$. I have the following question: Are there necessary and sufficient conditions on the CDF associated with $P$ (joint or ...
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Shouldn't the joint probability of 2 independent events be equal to zero?

If the joint probability is the intersection of 2 events, then shouldn't the joint probability of 2 independent events be zero since they don't intersect at all? I'm confused.
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How to derive joint CDF Gumbel distribution

If you have 3 random variables: $X$, $Y$, and $Z$ and they have independent Gumbel distribution. $A$, $B$ and $C$ are three discrete random variables that are functions of $X$, $Y$, and $Z$ as per the ...
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Changing the limits of integration when computing for marginal density [duplicate]

My question is based on this post. The question starts with $X \sim U(a, b)$ and $Y \sim U(a, X)$, and the answer computes the marginal density of $Y$. \begin{align} f(y) = \int_{-\infty}^{\infty} f(...
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Is “joint probability” assumption necessary for regression purposes?

I state beforehand that my question may sound odd and captious (and maybe it is). In regression theory basically we assume that explanatory variables and independent variable are joined togheter ...
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41 views

Finding joint support of $(XY,X/Y)$ where $(X,Y)$ has joint pdf $1/x^2y^2$ for $x,y\ge1$

$X$ and $Y$ are random variables with joint pdf $$\frac{1}{x^2y^2}\qquad,\, x\ge1,y\ge1$$ Set $$U=XY, V=X/Y$$ Explain why the joint range of $U$ and $V$ is given by: $$\{(u,v):v\in(0,1),u\ge1/v\} \...
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Calculating a probability based on a joint distribution between a Uniform random variable nested within a Uniform(0,1) random variable

Let $X_1 \sim Uniform(0,1)$, and $X_2 \sim Uniform(0, x_1)$, where $x_1$ is the realized value of $X_1$. Find $P(X_1 + X_2 \geq 1)$. I know that I need the joint distribution of $X_1$ and $X_2$. $...
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calculation of paramters needed for joint probability distribution?

Please correct me if I'm wrong. From my understanding, the number of entries in the above image is 7 because you need to calculate 7 and the 8th one can be done by 1-p. But I can't understand how ...
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47 views

To find the covariance given the joint probability density function.

Question: I was solving some question papers and got stuck in this problem. My problem: I know how to find "marginal probabilities" from a joint probability density function and also know how to ...
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Is there an analytical way of determining the probability for a specific outcome given a joint p.m.f.?

I'm looking for a way to determine the probability for a specific outcome based on (what I think should probably be) a joint probability mass function. I'll try to put into words my specific case: I ...
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How to validate if partitions of a training set have been generated by the same joint distribution?

Following Luxburg (2008) in a supervised learning problem: We do not make specific assumptions on the spaces $X$ or $Y$, but we do make an assumption on the mechanism which generates those training ...
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Should I use another random variable for the sum, if I work with random vectors?

Assume I have a simple random vector $(X, Y)$ with common distribution $P((0, 0))=1/6, P((1, 1))=1/6, P((3, 1))=1/4, P((0, 2))=1/6, P((1, 2))=1/4$, all others are zero. If I would like to argue ...
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how to find correlation coefficient when X and Y follows Poisson Distribution?

A bridge is examined for corrosion. It is believed that the corrosion on left side exist is poisson distributed with mean 3 and corrosion on right side is poisson distributed with mean 1.5+0.5X where ...
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What is the plot of the difference of joint distribution from independent joint distribution?

I wanted to plot the scatter-plot of many many data points for EDA but just got a big black blob. So I used density contours, and heatmaps aka 2d histograms. Better Example 2d histogram was not ...
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Uncorrelatedness + Joint Normality = Independence. Why? Intuition and mechanics

Two variables that are uncorrelated are not necessarily independent, as is simply exemplified by the fact that $X$ and $X^2$ are uncorrelated but not independent. However, two variables that are ...
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Loss function for comparing high-dimensional joint distributions

I'm synthesizing data trained from a source dataset, and am looking for a loss function to compare different data synthesis methods*. I have some ideas below, but each has drawbacks and none is very ...
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56 views

Can the marginal distributions of A,C and B,C be used to build joint distribution of A and B?

There are three random variables $A$, $B$ and $C$. If the variables $A$ and $B$ were independent, their marginal joint distribution would be given by $$ P(A,B) = P(A)P(B) $$ For example, given the ...
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How is exchangeability related to covariate shift?

I understand that exchangeability refers to the notion that the order of data in a sequence does not affect the joint distribution of that data. In a sense, the current data we possess is from the ...
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How does one write joint p.d.f., when the parameter (e.g. $\theta$) is a vector $\theta=(\theta_1,…,\theta_n)$?

How does one write joint p.d.f., when the parameter (e.g. $\theta$) is a vector $\theta=(\theta_1,...,\theta_n)$? And each $\theta_i$ is meant to correspond like $X_i \sim SomeDistr(\theta_i)$ Assume ...
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Limits of integration of a density function

My question is based on this post. In summary, $X \sim \text{Unif}(a,b)$ and $Y|X \sim \text{Unif}(a,X)$. Then the author does the following calculations: \begin{align} f(y) = \int_{-\infty}^{\infty} ...
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1answer
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Conditional Expectation of pdf

Wish to identify what I'm doing wrong when finding the $\operatorname E(X\mid Y=5)$ of the following: $$f(x, y)=\begin{cases} 1/6 & \text{if } 0<x<2, 0<y<6-3x \\ 0 & \text{...
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How to express the joint probability of two multivariate distributions over the same domain

I am working with multivariate normal distributions in scipy, but I have a question which is more statistics oriented: Say I have a multivariate distribution from two discrete random variables $X1$ ...
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1answer
27 views

Expected vs observed frequency of two events at the same time

I'll first give an example and afterwards a more formal definition of my problem. Example: Let's assume I'm looking at balls with two properties: their color can be black or white, their weight ...
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1answer
38 views

Showing indepedence of two random variables when $p(x,y) = p(x) \cdot p(y)$ except a constant factor?

During a course I attend at university, I encountered the following question: Given is a probability distribution: $$p(x,y) = \lambda \eta \cdot \exp(-\lambda x - \eta y) $$ supported on $\mathbb{R}...
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Determine the joint pdf of two new variables

I'm working on the following problem: and here's my attempt at a solution: From what I understand, $0<=x<=y<=1$ can be separated into $0<=x<=y$ and $x<=y<=1$, so when I plug in ...
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2answers
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Variance being negative

Let $X$ and $Y$ have joint pdf such that $$f(x,y) = 3e^{-3x-y}, 0 < x< \infty, 0< y< \infty.$$ (a) Show that $X$ and $Y$ are independent. (b) Calculuate $Var(X)$. In ...
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Joint probability distribution of geometric distribution

Let $X$ and $Y$ be independent and identically distributed $(i.i.d.)$ r.v.’s, each having the probability distribution, $p(k) = (1 − λ)λ^k$; $k = 0,1,...$ where $λ :(0; 1)$ is a constant. Define $U = ...
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When are correlated Normal random variables multivariate Normal?

I know that there are many example of correlated normal random variables which are not jointly (multivariate) normal. However, are there conditions which state when correlated normal random variables ...
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55 views

Joint pdf from joint cdf in R

I have a big matrix of data where for each element (column) I have a certain number of values. Using this data I computed, using the Emcdf library the Empirical Joint CDF for every couple of elements. ...
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4answers
254 views

Joint probability of multivariate normal distributions with missing dimensions

Suppose I conduct two experiments, each measuring a subset of possible parameters. From experiment #1 I measure two parameters and estimate the multivariate normal distribution $$ \mathbf{X}_1=\left [...
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1answer
39 views

Applying assumptions about marginal and conditional PDFs

We are given $0 < x_2 < x_1 < 1$. What assumptions can you make about $f_1(x_1)$ and $f_{2|1}(x_2|x_1)$? I know that $f(x_1) f_{2|1}(x_2|x_1) = \frac{1}{x_1}$. I know the expression can be ...
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Is it possible to find the joint distribution of a random vector if only the distribution of scalar many-to-one transformation is known? [duplicate]

Theoretical Exercise: I'd like to derive the joint distribution $p_{\boldsymbol{X}}$ of a random vector $\boldsymbol{X} \in \mathbb{R}^K$ if only the distribution of a scalar many-to-one ...
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Jensen–Shannon divergence relation between marginal and joint

I have the following distributions- Joints: $p_1$(x,y) = $\rho_1$(x) $q_1$(y|x), $p_2$(x,y) = $\rho_2$(x) $q_2$(y|x) And the marginals: $\rho_1$(x), $\rho_2$(x). Is it possible to provide any ...
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Distribution of maximum variance explained by 1 variable

Say I do principal component analysis on $n$ variables, and I sort the fractions of variance explained to find the largest. What is the probability distribution for this figure? For context I just ...
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1answer
106 views

Joint probability of two distributions

If I have one random variable that represents hours worked per job X~exponential($\theta$). I have another random variable that represents how many jobs obtained per month Y~Poisson($\lambda$). Using ...
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Predictive Posterior Distribution of Normal Distribution with Unknown Mean and Variance

Suppose that $x_{i}|\mu,\sigma^{2} \sim \mathcal{N}(\mu,\sigma^{2})$ for $i = 1,...n$. Assume that the assigned prior distributions are $\mu$ ~ $\mathcal{N}$($\mu_{0}$, $\sigma^{2}_{0}$) and $\tau \...
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1answer
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how to get joint pdf of mixed random variables

I would like to know how the joint probability density function $p(b,r,\sigma^2)$ can be calculated for the following graph. Random variable $b$ is a latent binary variable, and random variable $\...
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1answer
19 views

Joint densities and conditional independece

Let us assume the joint density $p(x,y,z)$ is factorized as $p(y)p(z|y)p(x|z)$. Hence, $x \perp y|z$. Now, the posterior distribution of z is: $p(z|x,y)=\frac{p(x,y,z)}{p(x,y)}$, where $p(x,y)=\int p(...
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1answer
46 views

Probability Dependent

Okay so for the following question (d), why is the probabily calculated by P(man)+P(hip replacement)-P(man and hip replacement)=110/200+80/200-57/110 rather than 110/200+80/200-57/200. Isn't the total ...
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marginalization of joint distributions

I am trying to understand the following sentence, section 2.2, in this paper: "...it is required that the joint mode $p(x,z,a)$ gives back the original $p(x,z)$ under marginalization over $a$, thus $...
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25 views

Prove that a multivariate density is valid

My question is quite simple. How to prove that a function $f(x_1,x_2,..., x_n)$ is a valid density? I mean, aside of the fact that must integrate to 1, and it must be positive, do i have to prove ...
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What's the velocity and displacement of a free particle?

I'm reading Ornstein and Uhlenbeck (1930). They calculate the velocity of a free particle at time t given an initial velocity at time zero to be normally distributed. They also calculate the ...
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1answer
37 views

optimal subset / joint distribution prediction with machine learning

How can I find the optimal subset of classes for a given entity? For context, say that we have some customers and data about these customers transactions, and a set of possible products to advertise ...
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2answers
29 views

Joint CDF from piecewise marginal

Let X be a positive r.v. with density $f$ with $f(x) > 0~ \forall~ x>0$. Let Y be a r.v. that is equal to X if $X \leq 5$ and $X^3$ if $X > 5$. I am self-studying probability, and I'm quite ...
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1answer
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How do you compute the P(x>y) for a joint density function in R?

I'm trying to understand the Bayesian AB testing process more thoroughly. If I have two tests such that the posteriors are: $$x\sim Beta(\alpha_1, \beta_1)$$ $$y\sim Beta(\alpha_2, \beta_2)$$ Where $...
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Reference for matrix-variate($n\times m$ random matrix) normal distribution

I'm looking for a document with some few pages with the basic properties of the Matrix-Variate Normal distribution, with an applied perspective. Is there such document? If not, what's the most similar ...
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2answers
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Is it possible to derive joint probabilities from marginals with assumptions about the conditionals?

I understand the title is too generic. I tried to look for similar questions and although there were a few that were seemingly about the same issue, either they provided answers in the negative or had ...
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0answers
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Given a joint probability distribution, how to estimate probability of the effect conditional to individual causes?

Given a joint probability distribution of an effect over several causes, what techniques would be sensible to estimate a conditional probability for the effect, conditional to individual causes? In ...
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1answer
45 views

Proof the joint bivariate cumulative distribution function

I would like to proof the expression ($P[X>x, Y>y]$) for two continuous random variables $X$ and $Y$. $P[X>x, Y>y]$ = $1- P[X \leq x, Y \leq y]$ (From the definition of probability). (...