If the above is not satisfied for all (x,y) then X and Y are dependent.
Note X and Y will be independent if every entry in the joint probability table is the product of the corresponding row and column marginal probabilities.
Ex. 9 Suppose the joint probability mass function of X and Y is given in the following table:
Y
X
0 100 200
100
250
0.20 0.10 0.20
0.05 0.15 0.30
Are X and Y independent? NO
Ex. 10 Suppose that the lifetimes of two components are independent of one another and that the first lifetime x1 has an exponential distribution with parameter while the second, x2, has an exponential distribution with parameter . Let = 1/1000 and = 1/1200 so that the expected lifetimes are 1000 hours and 1200 hours, respectively.
Find the joint pdf.
Find the probability that both component lifetimes are at least 1500 hours.
while the second, x2, has an exponential distribution with parameter 
. Let