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?
Answer:
Step1 of 5:
Given data states that a steel manufacturer testing a new additive, for manufacturing an alloy of steel.
The joint probability mass function of tensile strength (y) and additive concentration (x).

Tensile strength 

Concentration of additive 
100 
150 
200 
0.02 
0.05 
0.06 
0.11 
0.04 
0.01 
0.08 
0.10 
0.06 
0.04 
0.08 
0.17 
0.08 
0.04 
0.14 
0.12 
Step2 of 5:
a). Here we need to find the marginal probability mass functions for X and Y.

Tensile strength (y) 

Concentration of additive (x) 
100 
150 
200 

0.02 
0.05 
0.06 
0.11 
0.22 
0.04 
0.01 
0.08 
0.10 
0.19 
0.06 
0.04 
0.08 
0.17 
0.29 
0.08 
0.04 
0.14 
0.12 
0.3 
0.14 
0.36 
0.5 

The marginal probability mass function is found by summing along the rows of the joint probability mass function.
For additive concentration (x): , , ,
= 0 for
The marginal probability mass function is found by summing along the columns of the joint probability mass function.
For tensile strength (y):
= 0 for
Step2 of 5:
b). The aim is to find, X and Y are independent or not.
In this case, X and Y are not independent.
For example, but
= 0.0308.
Therefore,
So, X and Y are not independent.
Step3 of 5:
c). Here we need to find the probability that its strength is 150 or more.
We have a specimen has an additive concentration of 0.04.
We use the conditional probability, that is
= 0.9474
Therefore, the probability that specimen strength is 150 or more is 0.9474.
Step4 of 5:
d). To find the probability that its tensile strength is greater than 125.
We have a specimen has an additive concentration of 0.08.
We use the conditional probability, that is
= 0.867
Therefore, the probability that its tensile strength is greater than 125 is 0.867.
Step5 of 5:
e). Here we need to find, what additive concentration should be used to make the probability of meeting this specification the greatest.
The tensile strength is greater than 175 if Y