## D. Joint Distributions

Let X and Y be two random variables defined on the same sample space.

#### Example:

Sample Space
X
amount of outside study time put in by student (per day)
Y
overall average marks of the student

#### The Joint Cumulative Distribution of X and Y is defined as F(x,y) = P(X < = x & Y < = y)

Example (continued)
F(24,100) = 1
all students study less than 24 hours a day and all have averages less than or equal to 100
F(0,0) = 0
no students study less than 0 hours and have averages less than or equal to 0.

#### The marginal density function for X is

Notice that the marginal density is the same as the ordinary density for the single randome variable. The other random variable Y is integrated out (doesn't matter what value it takes on).

#### X and Y are (statistically) Independent if and only if for all values of x and y

In words: if the joint density factors into the product of the two marginal densities then X and Y are said to be independent.

#### Examples of joint expectations:

1. The mean of one of the random variables can be written using the joint density function:

This is the case where g(x,y) = y.
2. Covariance is defined as E[ (X-E[X])(Y-E[Y]) ]
Covariance extends The notion of variance to two dimensions. Its value measures the linear relationship between r.v. Positive covariance indicates that when X is above its mean then Y tends to be above its mean as well - positive statistical relationship. Negative covariance indicates that when X is above its mean Y tends to be below its mean - negative statistical relationship
3. Correlation between two random variables ( ) is defined as

( is a common synonym for corr).

#### Properties of Covariance

1. X and Y independent Cov[X,Y] = 0
2. Cov[X,Y]
3. If X and Y are normal random variables then Cov[X,Y] = 0 X and Y independent.
Note that Cov[X,2Y] = 2Cov[X,Y], so a simple linear change in the random variables will increase or decrease the size of their covariances. Correlation avoids this scaling effect.

#### Properties of Correlation

1. Cov[X,Y] = 0 corr[X,Y] = 0 (Zero covariance is the same as zero correlation)
2. If X and Y are linear functions of each other, then |corr[X,Y]| = 1.
3. corr[aX,bY] = corr[X,Y]
The absolute value of corr[X,Y] measures the degree of linear relationship between them. The sign of corr[X,Y] indicates whether there is a positive or negative relationship between X and Y. Unlike covariance, correlation is unaffected by the scaling of the random variables by constants.

#### Example of Joint Distribution

Let X take on the values 1, 2
Let Y take on the values 0,1

f (x)
f(x,y)    0         1          X
--------------------
1     |  0.1    |   0.3    |   0.4
|_________|__________|
|         |          |
2     |  0.4    |   0.2    |   0.6
---------------------
f (y)     0.5        0.5        1.00
Y

Confirm the following
E[X] = 1.5
E[Y] = 0.4
Var(X) = 0.25
Var(Y) = 0.24
X and Y are not independent
E[X | Y=0 ] = 1.8

### You can get as much practice as you want with this kind of example by running the tutorial for week 2.

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