Jointly distributed Random variables Multivariate distributions Discrete Random

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Jointly distributed Random variables Multivariate distributions

Jointly distributed Random variables Multivariate distributions

Discrete Random Variables

Discrete Random Variables

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

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

Continuous Random Variables

Continuous Random Variables

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

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

If then defines a surface over the x – y plane

Multiple Integration

Multiple Integration

A f(x, y) A

A f(x, y) A

If the region A = {(x, y)| a ≤ x ≤ b, c ≤

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

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

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

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

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

Example: Compute Now

The same quantity can be computed by reversing the order of integration

The same quantity can be computed by reversing the order of integration

Integration over non rectangular regions

Integration over non rectangular regions

Suppose the region A is defined as follows A = {(x, y)| a(y) ≤

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 ≤

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

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

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

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

and

and

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

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

Hence

Continuous Random Variables

Continuous Random Variables

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

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

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

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

The bivariate Normal distribution

Let where This distribution is called the bivariate Normal distribution. The parameters are m

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

Surface Plots of the bivariate Normal distribution

Note: is constant when is constant. This is true when x 1, x 2

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 and Conditional distributions

Marginal distributions for the Bivariate Normal distribution Recall the definition of marginal distributions for

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

Proof: The marginal distributions of x 2 is where

Now:

Now:

Hence Also and

Hence Also and

Finally

Finally

and

and

Summarizing where and

Summarizing where and

Thus

Thus

Thus the marginal distribution of x 2 is Normal with mean m 2 and

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

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

Proof

where and Hence Thus the conditional distribution of x 2 given x 1 is

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 marginal distributions

Bivariate Normal Distribution with conditional distribution

Bivariate Normal Distribution with conditional distribution

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

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