All Questions
Tagged with density-function marginal-distribution
35 questions
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Do marginal density functions derived from a joint pdf always integrate to 1 (are they valid pdf's)?
If I have a joint pdf of multiple random variables, say 3 for simplicity, $f_{X,Y,Z}(x,y,z)$, is it true that the marginal density functions derived from that joint probability distribution ( $f_{X}(x)...
- 11
3
votes
1
answer
281
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Two dimensional random variable with uniform marginal probability density functions [duplicate]
I have access to some data for two variables - let's call them x and y.
In particular, I have the distribution of data separately per each variable, something that allows me to estimate the marginal ...
1
vote
0
answers
105
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Completing the square and marginalizing a multivariate Gaussian [closed]
Edit: This question has been closed for being unrelated although I see similar questions posted here with the same objective, yet not with enough detailed answers or not exactly what I am looking for (...
- 31
3
votes
1
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177
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Calculate expected values E(x) & E(y) & variance of x & y of joint PDF, which was previously transformed from Polar to Cartesian
Given two independently uniform distributed random variables angle $\theta \in [0,2\pi]$ and radius $r \in [0,1]$.
I obtain for the joint density function with polar coordinates: $$ f_{r,\theta}(r,\...
- 33
0
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0
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158
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Integrating the product of two Gaussians with different dimensions
I'm relatively unfamiliar with sort of integration calculus, so apologies in advance for any notation issues.
Given parameters of some linear-regression model $\mathbf{a}=(a_1,a_2,a_3)$, and ...
4
votes
2
answers
373
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What is the copula of a variable with itself?
In Sklar's theorem for joint probability functions,
$$f(x,y) = c(F_X(x), F_Y(y)) \cdot f(x) f(y)$$
the copula is $c(\cdot)$ of variables $X$ and $Y$, while $f(\cdot)$ are their marginal distributions.
...
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1
vote
1
answer
46
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Marginal distribution
A loss distribution has PDF - $f(x) = 1/x^2$, for $x > 1$
An insurer finds that the time in hours it takes to process a loss amount x has a uniform distribution on the interval $(\sqrt x, 2\sqrt x)$...
3
votes
1
answer
102
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Finding P(a< u(X,Y) <b) given a rectangular support
>The continuous variables X and Y have the following joint pdf $f(x,y) = x + y, 0<x,y<1.$
Determine $P(0.5<X+Y<1.5)$.
I know that the support of x and y is rectangular, hence they are ...
- 33
1
vote
0
answers
47
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Integrate out the binary indicator variable in a two-sample ANOVA
I have two sets of data, A and B, that have unequal sizes, and I want to compare their means. The standard approach would be to do a t-test. Getting a little more sophisticated, we can think of that t-...
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3
votes
1
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166
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Marginal likelihood of implicit model
In the introduction of "Implicit Maximum Likelihood Estimation" (Li et al., 2018), implicit models are defined as the deterministic parameterized transformation $T_\theta(\cdot)$ of an analytic ...
1
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0
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37
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Conditional and density probability (normal distribution)
I am trying to solve the following problem:
Suppose that $\mu\sim N(1,4)$ and $Y|\mu\sim N(\mu,1)$. Show that:
$$\begin{bmatrix}Y \\ \mu \end{bmatrix} \sim N\bigg(\begin{bmatrix}1 \\ 1 \end{bmatrix},...
2
votes
1
answer
224
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Find conditional pdf given joint
Let the joint pdf of $X$ and $Y$ be $f(x,y) = 12e^{-4x-3y}, x>0, y>0$.
What is the marginal cdf of $X$? of $Y$?
Am I just supposed to integrate f(x,y) with respect to $x$ or $y$ to get the ...
- 125
1
vote
0
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18
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What does this assumption mean regarding equal marginal densities?
Suppose that we have a random variable $\epsilon$ with density $q(\epsilon)$ and $w = t(\theta, \epsilon)$, where $t$ is a deterministic function of a constant $\theta$ and random variable $\epsilon$. ...
- 286
1
vote
1
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498
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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:
$...
- 286
0
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1
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38
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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 $P(Y>2X)$,
...
- 115
3
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1
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9k
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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 ...
- 115
4
votes
2
answers
360
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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 the marginal pdf of $f_Y(y)$ but I am stuck.
8
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1
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2k
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The distribution of the initial point of an AR process
Consider a stochastic process $\{X_t, t = 1, 2, \ldots\}$ following the model $$X_t = \alpha X_{t-1} + e_t,$$ where $e_t \thicksim f$.
Can I say that the distribution of the initial point, $X_1$, is ...
- 333
2
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1
answer
295
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Solving a marginalization integral involving exponential distributions
I'm trying to solve a marginalization integral
\begin{equation}
\int p(y,w) dw
\end{equation}
in order to compute the density $p(y)$.
I assumed the following model:
\begin{equation}
y = (u+w)^2 + v
\...
- 324
4
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0
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728
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Constructing a joint distribution from pairwise bivariate marginal distributions?
It's fairly well-known that given univariate distribution functions $F_X, F_Y, F_Z$, one can construct the joint distribution $F_{(X, Y, Z)}(x, y, z) = C(F_{X}(x), F_{Y}(y), F_{Z}(z))$, where $C$ is ...
- 385
2
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3
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793
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Finding Marginal pmf
I am studying for an exam and have come across this problem:
Let the random variables $X$ and $Y$ have the joint pmf:
$f_{XY}(x,y)={2\over{n(n+1)}}$ for $y=1, . . . , x$; $x=1, . . . , n$
Find the ...
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2
votes
0
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145
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Sum of Correlated Empirical pdf via Gaussian Copula
I'm new to R.
My goal is to calculate and plot the probability density function of the sum of 3 correlated empirical random variables (X1+X2+X3), given the correlation matrix.
I want to aggregate the ...
2
votes
0
answers
217
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pdf of sum of squares error
If $Y_i$ (i=1,...,n) are iid N($\mu$,$\sigma^2$), how would I calculate the marginal pdf of the SSE?
On Wikipedia, I saw that $\sum$($Y_i$-$\bar{Y}$)$^2$ ~ $\sigma^2$$\chi$$^2$(n-1).
Any help would ...
- 21
2
votes
1
answer
283
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Is it possible to obtain joint PDF of set of variables given marginal PDF of each variable?
Say $f(X_1)$, $f(X_2)$, $f(X_3)$, $f(X_4)$ are the empirical marginal PDFs of random variables $X_1$, $X_2$ , $X_3$, $X_4$. Also given is correlation between each pair of variables $X_1$, $X_2$ , $X_3$...
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3
votes
1
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2k
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Marginal density and conditional density from joint density [duplicate]
I am having trouble understanding how to solve this when the variables are not discrete.
Let the simultaneous density of the non-discrete stochastic variables (X,Y) be
I am then supposed to find ...
10
votes
1
answer
2k
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Marginal probability function of the Dirichlet-Multinomial distribution
I can't seem to find a written out derivation for the marginal probability function of the compound Dirichlet-Multinomial distribution, though the mean and variance/covariance of the margins seem to ...
- 101
1
vote
1
answer
72
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Probability Density Functions of a Worker and his Waking up routine
The Problem: A worker wakes at 6 am and lies in bed for up to 2 hours. Upon rising it takes him an hour to shower and prepare which is preceded by him doing whatever he pleases. He never leaves for ...
- 21
0
votes
1
answer
405
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Consistency of density estimation under marginalization
Let $(x_1,y_1),\ldots,(x_n,y_n)$ be samples from some unknown distribution $p(X,Y)$ and $\hat{p}(X,Y)$, $\hat{p}(Y)$ density estimates of the joint and marginal distributions (i.e., for the estimation ...
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4
votes
1
answer
1k
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marginal conditional distribution from MCMC output [duplicate]
I have a MCMC sampler that targets $$\mathbb{P}(U_1,U_2,...U_n \mid G(U) \leq 0)$$ where $U=(U_1,U_2,...U_n)^T$. I realize now I am more interested in estimating the conditional density $$p_k = p(u_k \...
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4
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2
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1k
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Interpretation of cartesian product of the support of marginal distribution
Suppose we have a multivariate data set, $s = (s_1, s_2, ... s_p)$ and each $s_i$ is distributed with a distribution that has finite support (we'll call each $s_i$ a "source"). Let us ...
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1
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2
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Marginal and joint distributions of linear combinations of random vectors
Let $X_1,...,X_4$ be independent $N_p(μ,Σ)$ random vectors. Let $V_1,V_2$ be such that
$$V_1=(1/4)X_1-(1/4)X_2+(1/4)X_3-(1/4)X_4 $$
$$V_2=(1/4)X_1+(1/4)X_2-(1/4)X_3-(1/4)X_4 $$
I need to find the ...
- 429
1
vote
0
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926
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Marginal distribution of a function of order statistics
From the joint distribution of any two order statistics, say $Y_j$ and $Y_k$, $j<k$ I would like to derive the distribution of $Z=F(Y_k)-F(Y_j)$.
The initial pdf is:
$$f_{Y_j,Y_k} (y_j,y_k) =\...
- 21.1k
5
votes
1
answer
316
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Calculating the marginals given the conditional distributions
Suppose $X$ and $Y$ have conditional distributions given by:
\begin{align}
f(x|y)&\propto ye^{-yx}\;\;\text{for}\;\;0<x<B<\infty\\
g(y|x)&\propto xe^{-xy}\;\;\text{for}\;\;0<y<B&...
- 1,463
6
votes
1
answer
391
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Question about a marginal distribution
If I observe the following:
$X \sim N(\mu_x,\sigma^2_x)$
$Y|X=x \sim N(x,\sigma^2_y)$
My objective is to calculate the marginal distribution of $Y$.
(Since the variance term does not address some ...
- 1,113
10
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2
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11k
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Problem calculating joint and marginal distribution of two uniform distributions
Suppose we have random variable $X_1$ distributed as $U[0,1]$ and $X_2$ distributed as $U[0,X_1]$, where $U[a,b]$ means uniform distribution in interval $[a,b]$.
I was able to compute joint pdf of $(...
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