Joint Distribution

7/28/2019 Joint Distribution

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Two Random Variables

W&W, Chapter 5


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Joint Distributions

So far we have been talking about the

probability of a single variable, or a variable

conditional on another.We often want to determine the joint probability

of two variables, such as X and Y.

Suppose we are able to determine the following

information for education (X) and age (Y) for

all U.S. citizens based on the census.


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Joint Distributions

Education (X) Age (Y):

25-35

30

Age: 35-

55

45

Age: 55-

100

70

None 0 .01 .02 .05

Primary 1 .03 .06 .10

Secondary 2 .18 .21 .15

College 3 .07 .08 .04


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Joint Distributions

Each cell is the relative frequency (f/N).

We can define the joint probabilitydistribution as:

p(x,y) = Pr(X=x and Y=y)

Example: what is the probability of gettinga 30 year old college graduate?


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Joint Distributions

p(x,y) = Pr(X=3 and Y=30)

= .07

We can see that:

p(x) = y p(x,y)

p(x=1) = .03 + .06 + .10 = .19


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Marginal Probability

We call this the marginal probability

because it is calculated by summing

across rows or columns and is thusreported in the margins of the table.

We can calculate this for our entire table.


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Marginal Probability Distribution

Education

(X)

Age (Y):

30 45 70

p(x)

None: 0 .01 .02 .05 .08

Primary: 1 .03 .06 .10 .19

Secondary:

2

.18 .21 .15 .54

College: 3 .07 .08 .04 .19

p(y) .29 .37 .34 1


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Independence

Two random variables X and Y are

independent if the events (X=x) and

(Y=y) are independent, or:p(x,y) = p(x)p(y) for all x and y

Note that this is similar to Event E is

independent of F if:

Pr(E and F) = Pr(E)Pr(F) Eq. 3-21


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Example

Are education and age independent?Start with the upper left hand cell:

p(x,y) = .01p(x) = .08

p(y) = .29

We can see they are not independentbecause (.08)(.29)=.0232, which is notequal to .01.


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Independence

In a table like this, if X and Y are

independent, then the rows of the table

p(x,y) will be proportional and so will thecolumns (see Example 5-1, page 158).


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Covariance

It is useful to know how two variables varytogether, or how they co-vary. We

begin with the familiar concept ofvariance (E is expectation).

2 = E(x- )2 = (x- )2 p(x)

X,Y = Covariance of X and Y= E(X - X)(Y - Y)

= (X - X)(Y - Y)p(x,y)


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Covariance

Lets calculate the covariance for education (X)and age (Y).

First we need to calculate the mean for X andY:X = xp(x) = (0)(.08)+(1)(.19)+(2)(.54)+(3)(.19)=1.84

Y = yp(y) = (30)(.29)+(45)(.37)+(70)(.34)=49.15

Now calculate each value in the table minus itsmean (for X and Y), multiplied by the jointprobability!


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Covariance

X,Y = (X - X)(Y - Y)p(x,y)

= (0-1.84)(30-49.15)(.01) +

(0-1.84)(45-49.15)(.02) + (0-1.84)(70-49.15)(.05) +

(1-1.84)(30-49.15)(.03) + (1-1.84)(45-49.15)(.06) +

(1-1.84)(70-49.15)(.10) + (2-1.84)(30-49.15)(.18) +

(2-1.84)(45-49.15)(.21) + (2-1.84)(70-49.15)(.15) +(3-1.84)(30-49.15)(.07) + (3-1.84)(45-49.15)(.08) +

(3-1.84)(70-49.15)(.04) = -3.636


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Covariance and Independence

If X and Y are independent, then they areuncorrelated, or their covariance is zero:

X,Y = 0

The value for covariance depends on the unitsin which X and Y are measured. If X, for

example, were measured in inches instead offeet, each X deviation and hence X,Y itselfwould increase by 12 times.


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Correlation

We can calculate the correlation instead:

= X,Y

X Y

Correlation is independent of the scale it is

measured in, and is always bounded:

-1 1


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Correlation

A perfect positive correlation (=1); all x,y coordinate

points will fall on a straight line with positive slope.

A perfect negative correlation (=-1); all x,y coordinate

points will fall on a straight line with negative slope.

A correlation of zero indicates no relationship between

X and Y (or independence!).

Positive correlations (as X increases, Y increases)

Negative correlations (as X increases, Y decreases)


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Example of Correlation

Calculate the correlation between

education and age:

= X,Y = -3.636

X Y (.8212)(16.14)

= -0.2743


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Interpretation

There is a weak, negative correlation

between education and age, which

means that older people have lesseducation.

Later on we will learn how to conduct a

hypothesis test to determine ifissignificantly different from zero.

Joint Distribution

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