Jointly distributed Random variables Multivariate distributions Discrete Random

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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

If then defines a surface over the x – y plane

Multiple Integration

A f(x, y) A

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

To evaluate A First evaluate the inner integral Then evaluate the outer integral

y d dy y c a b x = area under surface above the line where y is constant Infinitesimal volume under surface above the line where y is constant

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

y dx d c a x b x = area under surface above the line where x is constant Infinitesimal volume under surface above the line where x is constant

Example: Compute Now

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} y d c Then A a(y) b(y) x

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

In general the region A can be partitioned into regions of either type y A 2 A 1 A 3 A A 4 x

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} y (0, 1) x+y=1 (1, 0) x

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

and

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

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 m 1, m 2 , s 1, s 2 and r.

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 m 1, m 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 xi is Normal with mean mi and standard deviation si.

Proof: The marginal distributions of x 2 is where

Now:

Hence Also and

Finally

and

Summarizing where and

Thus

Thus the marginal distribution of x 2 is Normal with mean m 2 and standard deviation s 2. Similarly the marginal distribution of x 1 is Normal with mean m 1 and standard deviation s 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 xi given xj is Normal with: mean standard deviation and

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

x 2 ( m 1, m 2) Major axis of ellipses Regression to the mean x 1