Bivariate probability distributions describe the relationship between two random variables. There are both discrete and continuous cases. For discrete variables, the relationship is defined by a joint probability mass function p(y1,y2). For continuous variables, it is defined by a joint probability density function f(y1,y2). Both must be non-negative and integrate/sum to 1. The cumulative distribution function F(y1,y2) gives the probability that both variables are less than or equal to values y1 and y2.

1. Bivariate Probability
Distributions

2. Bivariate Probability Distributions
• Joint Discrete Random Variables
• Joint cumulative distribution function (cdf)
F(y1,y2)
• Joint probability mass function (pmf) p(y1,y2)
• Joint Continuous Random Variables
• Joint cumulative distribution function (cdf)
F(y1,y2)
• Joint probability density function (pdf) p(y1,y2)

3. Joint probability mass function (pmf)
• Let Y1 and Y2 be discrete random
variables. The joint (or bivariate) probability
mass distribution for Y1 and Y2 is given by
• p(y1,y2)P(Y1y1, Y2y2)
• where -??y1 ??, -??y2 ??
• The function p(y1,y2) is called the joint
probability mass function (pmf) of Y1 and Y2

4. Properties of pmf
• Theorem 5.1 If Y1 and Y2 be discrete random
variables with joint probability mass
distribution p(y1,y2), then
• (1) p(y1,y2) ?0 for all y1,y2
• (2) ?y1 ?y2 p(y1,y2) 1, where the sum is over
all values (y1,y2) that are assigned nonzero
probabilities.

5. Cumulative distribution function( cdf)
• Definition 5.2 For any random variables Y1 and
Y2, the joint (bivariate) distribution function
(or cumulative distribution function, cdf)
F(y1,y2) is given by
• F(y1,y2)P(Y1?y1, Y2?y2)
• where -??y1 ??, -??y2 ??

6. Joint Probability Density Function (pdf)
• Let Y1 and Y2 be continuous
random variables with joint distribution function
F(y1,y2). If there exists a nonnegative function
f(y1,y2) such that
• then Y1 and Y2 are said to be jointly continuous
random variables. The function f(y1,y2) is called
the joint probability density function (pdf).
•

7. Properties of cdf
• If Y1 and Y2 are random variables
with joint cumulative distribution function
F(y1,y2), then
• (1) F(-?,-?)F(-?,y2)F(y1,-?)0
• (2) F(?, ?)1
• (3) If y1?y1 and y2?y2, then
• F(y1,y2)-F(y1,y2)-F(y1,y2)F(y1,y2) ?0

8. Properties of pdf
• If Y1 and Y2 are jointly continuous
random variables with a joint density function
(pdf) given by f(y1,y2), then
• (1) f(y1,y2) ?0 for all y1, y2