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/in2) 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?

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Tensile Strength Concentration of Additive 100 150 200 0.02 0.04 0.06 0.08 0.05 0.06 0.11 0.0 008 0.10 0.04 0.08 0.17 0.04 0.14 0.12