Introduction to Probability & Statistics Joint Distributions

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.

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

Discrete Bivariate

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

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

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.

Conditional Distribution

Conditional Distribution

Condition on Field Placement

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

Conditional; Engineering

Conditional

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

Marginal Distribution

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

Marginal Distribution

Marginal

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  ( , )

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 ( )

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)

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

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