1. Introduction to Probability & Statistics Joint Distributions

2. Discrete Bivariate
Suppose we track placement data for 1,000 recent graduates at a local university. Students are tracked by undergraduate major and are placed in one of three categories.

3. Discrete Bivariate
If we divide the number in each cell, by 1,000 we then have defined a discrete bivariate distribution.

4. Discrete Bivariate

5. Discrete Bivariate p x y X Y , ( )
Pr{ } = p x y , ( ) . = å 1

6. Conditional Distribution
Suppose that we wish to look at the distribution of students going to graduate school.

7. Conditional Distribution
We are now placing a condition on the sample space that we only want to look at students in graduate school. Since the total probability of students in graduate school is only 0.21, we must renormalize the conditional distribution of students in graduate school.

8. Conditional Distribution

9. Conditional Distribution

10. Condition on Field Placement

11. Conditioning on Major
Suppose, we wish to condition by major (y-axis) and look at placement for engineers only.

12. Conditional; Engineering

13. Conditional

14. Marginal Distribution
The marginal distribution for placement by major is just the sum of joint probabilities for each major.

15. Marginal Distribution

16. Marginal Distribution
We can find the marginal distribution by category in a similar fashion.

17. Marginal Distribution

18. Marginal

19. Bivariate Uniform  f x y dxdy ( , ) Pr{X< a, Y< b} = b a
fXY(x,y) x y b a
Pr{X< a, Y< b} = f x y
dxdy XY b a  ( , )

20. Bivariate Uniform   ( ) f x y dydx   f x dx y dy ( )
fXY(x,y) x y b a
For X,Y independent, fXY(x,y) = fX(x)fY(y) ( ) f x y
dydx X b a Y  
Pr{X< a, Y< b}   f x dx y dy X a Y b ( )

21. Conditional Distribution
fXY(x,y) x y
Lets take a slice out of fXY(x,y) a particular value of x.
The area under the response surface fXY(x,y) is just the
conditional probability of y for a specific value of x. Or
f(y|x) = fXY(x,y|x)

22. Marginal Distribution
fXY(x,y) x y
If we look at this for all values of x, we get f ( y )   f ( x , y ) dx Y x XY

23. Marginal Distribution
fXY(x,y) x y
Similarly, looking at the slice the other way ) ( f x y dy X XY ,  