In a piston assembly, the specifications for the clearance between piston rings and the cylinder wall are very tight. In a lot of assemblies, let X be the number with too little clearance and let Y be the number with too much clearance. The joint probability mass function of X and Y is given in the table below:
a. Find the marginal probability mass function of X.
b. Find the marginal probability mass function of Y.
c. Are X and Y independent? Explain.
d. Find 
and 
.
e. Find 
and 
.
f. Find Cov(X,Y).
g. Find 
.
(a) The marginal probability mass function 
is found by summing along the rows of the joint probability mass function.
= 0 if X ?0, 1, 2, or 3
(b) The marginal probability mass function 
is found by summing down the columns of the joint probability mass function. So 
= 0 if y ?0, 1, 2, or 3
(c) No, X and Y are not independent. For example, P(X = 0 and Y = 0) = 0.15, but P(X = 0)P(Y = 0) = (0.48)(0.34) = 0.1632 ? 0.15.
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