Jointly distributed Random variables Multivariate distributions

Discrete Random Variables

The joint probability function; p(x,y) = P[X = x, Y = y]

Continuous Random Variables

Definition: Two random variable are said to have joint probability density function f(x,y) if

Ifthen defines a surface over the x – y plane

Multiple Integration

A A f(x,y)

If the region A = {(x,y)| a ≤ x ≤ b, c ≤ y ≤ d} is a rectangular region with sides parallel to the coordinate axes: x y d c ab A f(x,y) Then

A f(x,y) To evaluate Then evaluate the outer integral First evaluate the inner integral

x y d c ab y f(x,y) = area under surface above the line where y is constant dy Infinitesimal volume under surface above the line where y is constant

A f(x,y) The same quantity can be calculated by integrating first with respect to y, than x. Then evaluate the outer integral First evaluate the inner integral

x y d c ab x f(x,y) = area under surface above the line where x is constant dx Infinitesimal volume under surface above the line where x is constant

f(x,y) Example: Compute Now

f(x,y) The same quantity can be computed by reversing the order of integration

Integration over non rectangular regions

Suppose the region A is defined as follows A = {(x,y)| a(y) ≤ x ≤ b(y), c ≤ y ≤ d} x y d c a(y)a(y) b(y)b(y) A Then

If the region A is defined as follows A = {(x,y)| a ≤ x ≤ b, c(x) ≤ y ≤ d(x) } x y b a d(x)d(x) c(x)c(x) A Then

In general the region A can be partitioned into regions of either type x y A1A1 A3A3 A4A4 A2A2 A

f(x,y) Example: Compute the volume under f(x,y) = x 2 y + xy 3 over the region A = {(x,y)| x + y ≤ 1, 0 ≤ x, 0 ≤ y} x y x + y = 1 (1, 0) (0, 1)

f(x,y) Integrating first with respect to x than y x y x + y = 1 (1, 0) (0, 1) (0, y) (1 - y, y) A

and

Now integrating first with respect to y than x x y x + y = 1 (1, 0) (0, 1) (x, 0) (x, 1 – x ) A

Hence

Continuous Random Variables

Definition: Two random variable are said to have joint probability density function f(x,y) if

Definition: Let X and Y denote two random variables with joint probability density function f(x,y) then the marginal density of X is the marginal density of Y is

Definition: Let X and Y denote two random variables with joint probability density function f(x,y) and marginal densities f X (x), f Y (y) then the conditional density of Y given X = x conditional density of X given Y = y

The bivariate Normal distribution

Let where This distribution is called the bivariate Normal distribution. The parameters are  1,  2,  1,  2 and 

Surface Plots of the bivariate Normal distribution

Note: is constant when is constant. This is true when x 1, x 2 lie on an ellipse centered at  1,  2.

Marginal and Conditional distributions

Marginal distributions for the Bivariate Normal distribution Recall the definition of marginal distributions for continuous random variables: and It can be shown that in the case of the bivariate normal distribution the marginal distribution of x i is Normal with mean  i and standard deviation  i.

The marginal distributions of x 2 is where Proof:

Now:

Hence Also and

Finally

and

Summarizing where and

Thus

Thus the marginal distribution of x 2 is Normal with mean  2 and standard deviation  2. Similarly the marginal distribution of x 1 is Normal with mean  1 and standard deviation  1.

Conditional distributions for the Bivariate Normal distribution Recall the definition of conditional distributions for continuous random variables: and It can be shown that in the case of the bivariate normal distribution the conditional distribution of x i given x j is Normal with: and mean standard deviation

Proof

where and Hence Thus the conditional distribution of x 2 given x 1 is Normal with: and mean standard deviation

Bivariate Normal Distribution with marginal distributions

Bivariate Normal Distribution with conditional distribution

(  1,  2 ) x2x2 x1x1 Regression Regression to the mean Major axis of ellipses