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

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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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3answers
66 views

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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0answers
22 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
217 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
36 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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0answers
15 views

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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0answers
29 views

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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0answers
11 views

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
40 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 ...
2
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0answers
65 views

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
28 views

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
45 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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0answers
23 views

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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1answer
22 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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0answers
4 views

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
32 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
25 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
82 views

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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1answer
57 views

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
106 views

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
9 views

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
34 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). (...
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0answers
13 views

symmetric marginal but asymmetric joint distribution contours [duplicate]

Let us say we have two continuous random variables, $X$ and $Y$ such that their pdfs $f(x)= f(-x)$ and $g(y)= g(-y)$ for all $x$ and $y$. In other words, $X$ and $Y$ have symmetric distributions ...
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0answers
17 views

Truncation and a composite distribution

Suppose $X$~$N(a,1)$ $Y|X$~$N(X,\sigma^2)$ Then what is $X|Y<0$ ~ ? and $E[X|Y<0]$ ~ ?
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0answers
37 views

Check whether a random sample comes from an elliptical distribution?

How can I check whether it is a reasonable assumption to say that a multivariate sample $x_1,...,x_n$ comes from an elliptical distribution, such as a normal distribution or a t-distribution? In the ...
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2answers
50 views

Convolution of a less typical distribution

$X_1$ and $X_2$ are independent and identically distributed (i.i.d) random variables defined on R+ each with pdf of the form $f_X(x) = \sqrt\frac{1}{2\pi x}exp[\frac{-x}{2}]\quad ,\quad x>0, \quad ...
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0answers
19 views

Copula and non-Copula models

I am working with copula-based models. Copula models allow to models the margins separately from the dependencies structures. However, non-copula models do not allow for such separation. My question ...
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1answer
175 views

Analytically solving sampling with or without replacement after Poisson/Negative binomial

Short version I am trying to analytically solve/approximate the composite likelihood that results from independent Poisson draws and further sampling with or without replacement (I don't really care ...
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1answer
32 views

Joint CDF of dependent random variables: is knowing covariance sufficient?

Let $X,Y$ be real-valued random variables, which are dependent. Want: Calculate $\mathbb{P}[\,\min\{X,Y\}\leqslant0\,]$ (without Monte Carlo) Know: I can compute (numerically) $F_X$ and $F_Y$ (the ...
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0answers
8 views

I have a data frame with 3 variables: Week | Count | Length. How can i get a joint pdf of count and length wrt to Week (time variable)

I tried using the auto.arima function in R but it only works with univariate time series. Can someone please help me with this? T The data-frame looks like: Week (1,2...92 | Count | Length I ...
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0answers
10 views

How to use Copulas to Combine Multivariate Conditional Probability with Univariate Conditional Probability?

This is sure to be an odd one, but here goes. I'm trying to estimate P(X|Y, Z) by the distributions of P(X|Y) and P(X|Z). I've thus far been trying to using copulas to achieve that aim, but I'm not ...
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1answer
23 views

Random generation of wealth with normal distribution of two parameters? [duplicate]

I want to randomly generate the wealth of a group of people, with two parameters: age and height. Basically (not necessarily realistic): Rule 1. the older a person (allow decimals), the higher the ...
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1answer
43 views

Computing joint entropy from marginal distributions

I have distributions of N random variables (supposed conditionally independent) consequently, the joint distribution is the multiplication of all the distributions. I want to compute the joint ...
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0answers
8 views

Joint estimation for same set of independent variables for different but corralated dependent variables

I am using cross country data to regress the different income deciles (Di) on the same macroeconomic indicators, which gives a set of equations of the form. D1= B1x + u1 D2= B2x + u2 .... D10= B10x + ...
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0answers
17 views

Variable substitution in joint probability density function [duplicate]

Given two continuous random variables $X$ and $Y$ (they can be dependent) and joint probability density function $f_{X,Y}(x,y)$. The question is how to find joint probability density function $f_{X,Z}(...
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1answer
152 views

Sufficient conditions for joint normality?

Suppose I have $n$ random variables $X_1,...,X_n$ such that $X_1 \sim \mathcal{N}(0,1)$ and the increments $X_i - X_{i-1} \sim \mathcal{N}(0,s)$ are independent. Are these conditions sufficient to ...
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0answers
40 views

Getting marginal distributions from a bivariate probability distribution function [duplicate]

I understand the basic principles of bivariate distributions and their marginal counterparts. I am stuck on a slightly more sophisticated question however. Given the following bivariate distribution: ...
0
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1answer
31 views

Construct joint distribution of $X,Y$ such that $E[X|Y=y,y\geq \bar{y}]$ is piecewise linear

Can one construct a joint density $f(x,y)$ such that the marginal distribution of $Y\sim~U[c,d]$, no restrictions on $X$ (it would be great that $X$ also has uniform distribution) as long as it has ...
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0answers
60 views

Estimating parameters and simulating a multivariate time series (O-U?) process

Say I have 5 time series that may or may not be correlated with each other. I also suspect they individually may be mean reverting. I'd like to 1) be able to test the hypothesis they are ...
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0answers
17 views

Creating a joint density from two asymmetric uniform marginals

Two RV \begin{split} X_1 & \overset{{i.i.d.}}{\sim}\mathcal{U}[0,1] \\ X_2 &\overset{{i.i.d.}}{\sim}\mathcal{U}[0.3,2.08] \end{split} I need to create a joint pdf from these two. Is copula ...
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0answers
29 views

Factorization of product distribution

I would like to obtain a factorization of the probability distribution of a random variable $Z$ which is the product of two independent random variables $\delta=\varepsilon X$ where $\varepsilon \sim ...
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0answers
24 views

Finding pivot from joint distribution and then 95% exact confidence interval

I am trying to find the 95% Exact confidence interval with the below scanned joint distribution and parameters. For the given joint pdf I believe it is exponential (location-scale). My approach is at ...
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1answer
85 views

Calculating joint probabilities from conditional probabilities, categorical variables

I want to calculate the joint probabilities P(X,Y), where X and Y are both categorical variables with 7 and 6 categories each. Given the conditional probabilities P(X|Y), is it possible to calculate ...
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0answers
123 views

Joint PDF and Joint CDF of a Discrete and Continuous random variables

Suppose we have a discrete random variable $X$ and a continuous random variable $Y$. I am trying to understand how one defines/ find the joint PDF and joint CDF of $X$ and $Y$. The joint CDF of $X$ ...
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1answer
29 views

Why is the marginal continuous pdf of X+Y in this form?

I read in a book that given a joint continuous pdf $g(x+y)$, for $y$ is: $$ \int_{0}^{\infty}g(x+y)dx = \int_{y}^{\infty}g(x)dx $$ I got stuck here, how did $\int_{y}^{\infty}g(x)dx $ come out?
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0answers
20 views

Let $\xi$ random vector and $\zeta^{x}:=x^{T}\xi$ random variable. Is it correct to say that $\int x^{T}y f_{\xi}(y)dy=\int tf_{\zeta^{x}}(t)dt$?

Let $\xi\in\mathbb{R}^{m}$ be a random vector with joint density function $f_{\xi}:\mathbb{R}^{m}\rightarrow \mathbb{R}$. Let $x\in \mathbb{R}^{x}$ be a nonzero vector. We consider the random ...
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0answers
14 views

Express E[AAB] in terms of marginal moments and E[AB] only

For a pair of generic random variables $(A, B)$, I am trying to find a way to express $$E[A^2B]$$ using only the marginal moments $$E[A^k], E[B^k], k=1,2,\ldots $$ and the second joint moment $$...
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0answers
9 views

Finding a reasonable threshold for binning undersampled classes

I have a large survey and want to estimate the joint probability distribution of a subset of the survey-variables. E.g. I have the categorical variables "age-class" and "income-class" and want to ...
2
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1answer
107 views

Why can Gibbs sampling outputs be used in Rao-Blackwellization?

I'm currently learning Chib (1995)'s method to calculate the marginal likelihood of a Bayesian model using Gibbs sampling outputs. I'm stuck in the Rao-Blackwellization step. Suppose $\mu$ and $\phi$...