A steel manufacturer is testing a new additive for manufacturing an alloy of steel. The joint probability mass function of tensile strength (in thousands of pounds/
) and additive concentration is
a. What are the marginal probability mass functions for X (additive concentration) and Y(tensile strength)?
b. Are X and Y independent? Explain.
c. Given that a specimen has an additive concentration of 0.04, what is the probability that its strength is 150 or more?
d. Given that a specimen has an additive concentration of 0.08, what is the probability that its tensile strength is greater than 125?
e. A certain application calls for the tensile strength to be 175 or more. What additive concentration should be used to make the probability of meeting this specification the greatest?
The marginal probability mass function 
(x) is found by summing along the rows of the joint probability mass function.
(a) For additive concentration (X): 
(0.02) = 0.22, (0.04) = 0.19, (0.06) = 0.29, (0.08) =0.30, and (x) = 0 for x?0.02, 0.04, 0.06, or 0.08.
For tensile strength (Y): The marginal probability mass function 
(y) is found by summing down the columns of the joint probability mass function. Therefore (100) = 0.14, (150) = 0.36, (200) = 0.50, and (y) = 0 for y ?100, 150, or 200.
(b) No, X and Y are not independent. For example P(X = 0.02 ? Y = 100) = 0.05, but P(X = 0.02)P(Y= 100) = (0.22)(0.14) = 0.0308.
(c)
(d)
(e) The tensile strength is greater than 175 if Y= 200. Now
The additive concentration should be 0.06.
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