Answer to Question #283481 in Statistics and Probability for Ruchi

Question

Explain the following - 1) two dimensional random variable, 2)marginal and conditional distributions, 3)bivarate normal distribution and it's properties

Answer 1

1)

 Let X and Y be two random variables defined on S. then the pair (X,Y) is called a Two – dimensional random variable. The value of (X,Y) at a point is given by the ordered pair of real numbers (X(s), Y(s)) = (x, y) where X(s) = x, Y(s) = y.

2)

Marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables.

Given two jointly distributed random variables  X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value x of X as a parameter.

3)

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. Visually, the bivariate normal distribution is a three-dimensional bell curve.

The bivariare distribution can be described in many different ways and as such, there isn’t a unified agreement for a succinct definition. Some of the more common ways to characterize it include:

• Random variables X & Y are bivariate normal if aX + bY has a normal distribution for all a,b∈R.
• X and Y are jointly normal if they can be expressed as X = aU + bV, and Y = cU + dV
• If a and b are non-zero constants, aX + bY has a normal distribution
• If X – aY and Y are independent and if Y – bx and X are independent for all a,b (such that ab ≠ 0 or 1), then (X,Y) has a normal distribution

PDF:

$P(x_1,x_2)=\frac{1}{2\pi \sigma_1 \sigma_2\sqrt{1-\rho^2}}exp(-\frac{z}{2(1-\rho^2)})$

where

$z=\frac{(x_1-\mu_1)^2}{\sigma_1^2}-\frac{2\rho (x_1-\mu_1)(x_2-\mu_2)}{\sigma_1 \sigma_2}+\frac{(x_2-\mu_2)^2}{\sigma_2^2}$

$\rho$ is correlation of x₁ and x₂