Answer to Question #316671 in Statistics and Probability for Dallas

Question

Suppose X and Y are jointly normal random variables. Briefly discuss bivariate Normal distribution. Your answer should include, but not limited to, the joint pdf of the bivariate normal distribution f(x, y), it's properties including, but not limited to, E(Y/X=x) and V(Y/X=x).

Answer 1

The “regular” normal distribution has one random variable; A bivariate normal distribution is made up of two independent random variables. The two variables in a bivariate normal are both are normally distributed, and they have a normal distribution when both are added together.

The joint probability density function (joint pdf) is a function used to characterize the probability distribution of a continuous random vector. It is a multivariate generalization of the probability density function (pdf), which characterizes the distribution.

X and Y are jointly normal random variables with parameters μ_X, σ²_X, μ_Y, σ²_Y, and ρ.

The joint probability density function (joint pdf) is a function used to characterize the probability distribution of a continuous random vector. It is a multivariate generalization of the probability density function (pdf), which characterizes the distribution of a continuous random variable.

Then, given X=x, Y is normally distributed with

E[Y|X=x]=$\mu_Y+\rho \sigma_Y \frac{x- \mu_X}{\sigma_X}$

$Var(Y|X=x)=(1-\rho^2)\sigma^2_Y$