Questions tagged [conditional-distribution]
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18 questions
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Maximum Likelihood Estimation for Pairs of Observations
I have $n$ pairs of observations $(x_i,y_i)$, where each $y_i$ is distributed according to $\text{Pois}(\theta x_i)$, and I wish to do a maximum likelihood estimation for $\theta$ only based on this ...
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Find marginal pdf of $Y$ if given pdf of $X$ and pdf of $Y\mid X$
If $X \sim \text{Poisson}(\mu)$ and $Y\mid X\sim U(X-0.5,X+0.5)$ and $-0.5<y<\infty$, find marginal pdf of $Y$.
I get that we need to find $f_X,_Y(x,y)$ by multiplying $f(y\mid x)\cdot f_X(x)$ ...
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BIvariate Normal and Conditional Expectation
I am working on a problem where I must show that the conditional distribution of Y given X follows the distribution with mean and variance shown below. In the previous question, we were given that X ...
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Splitting a random variable to give another random variable from the same family
Suppose you have a Poisson distributed random variable $N$ with expectation $\mu$. You then take a conditionally binomially distributed random variable $X$ where $X \sim \textrm{Bin}(N,p)$ for some $...
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Conditional distribution of $X_i|\overline{X}$
Suppose $x_1 \cdots x_n$ are iid random variables with normal distribution of mean $\mu$ and variance $1$, and $\overline{X}=\frac{1}{n}\sum_i^n x_i$
What is the conditional distribution of $\begin{...
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Conditional distribution $f(x|y)$ if $X$ and $Y$ are independent
Suppose we have two randome variables $X$ and $Y$, with joint distribution $f(x,y)$. $X$ and $Y$ are independent if and only if the marginal distribution of $X$ is the same as the conditional ...
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When do marginal/conditionally normals imply joint normal?
I know that marginal normals do not imply joint normal, as some examples in here gives. However, but I don't know of any theorems that talk about conditions under which marginal normals can imply ...
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Loss function for conditional variance?
Minimizing square loss results in predicting conditional means.
Minimizing absolute loss results in predicting conditional medians.
What loss function results in predicting conditional variances?
I ...
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How to express a likelihood function for the following regression?
I have asked this question in entirely different forms a number of times on StackExchange, to no avail. Between each question, I investigated the literature thoroughly, but I have yet to find a ...
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Finding the unconditional distribuition of a variable
Im solving Gibbs sampler related questions and Im trying to find the unconditional distribuition of $\beta | \lambda \sim N(0, \lambda^{-1}\Sigma)$ knowing that $\lambda \sim Gamma(\frac{\alpha}{2}\...
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Find the conditional distribution of $(x_2, x_3),$ given $x_1$ [duplicate]
If we let $x∼N_3(\mu, \Sigma)$ with $\mu^T=(\mu_1,\mu_2,\mu_3)$ and $\Sigma=\begin{pmatrix}\sigma ^2&\sigma ^2\rho &\sigma \:^2\rho \:\\ \sigma \:^2\rho \:&\sigma \:^2&\sigma \:^2\rho \...
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Strict stationarity in terms of conditional distributions
Let's start with the definition of a strictly stationary process: The process $\{X_t\}=\{X_1,X_2,X_3,X_4\}$ is strictly stationary if the joint distribution of the vector $(X_1,...,X_n)$ and the time ...
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How to Interpret Conditional Distribution
Considering the following contingency table:
We calculate the conditional distribution for the city Manchester:
Why do we need the conditional distribution and how we interpret the result?
Is ...
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Use Kernel Density Estimation for Prediction
I would like to use the KDE to predict future positions of surrounding vehicles. So given a set of data [I: some input features, O: future position] I learn the joint distribution of the input and ...
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Identity of ${{\mathit f}({\mathbf z} {\mid} {\mathbf x)}}$ and ${\mathit f}$($\mathbf {z}$) under normality - a peculiar case
I am a newbie to econometrics, so kindly excuse me if I sound too naive.
This is what Fumio Hayashi says on page 34 of "Econometrics":
Recall from probability theory that the normal distribution ...
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Gibbs sampling example of a bivariate normal with unknown correlation
I'm looking for an example of using Gibbs sampling with a bivariate normal, where the correlation parameter is not fixed or known. In other words, what is the conditional distribution of the ...
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How do I find the conditional distribution of a normal r. v. z, given that I know the sum of z and another normal r. v. x is greater than some value?
Suppose I have two independent normal random variables, $X$ and $Z$ with $\mu_x$, $\sigma^2_x$ and $\mu_z$, $\sigma^2_z$. Suppose I also know that $x+z\geq y$. How do I find the conditional ...
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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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