The zinc-phosphate coating onthreads of steel tubes used in oil and gas wells is

Question

The zinc-phosphate coating onthreads of steel tubes used in oil and gas wells is critical to their performance. To monitor thecoating process, an uncoated metal sampling with known outside are is weighed and treatedalong with the lot of tubing. This sample is then stripped and reweighed. From this it is possible to determine whether the correct amount of coating was applied to the tubing.Assume that the probability that a given lot is unacceptable is 0.05. Let X denote the numberof runs conducted to produce an unacceptable lot. Assume that the number of runs areindependent in the sense that the outcome of one run has no effect on that of any other.

a. What kind of discrete random variable is X and what is its parameter(s)? (5 pts each)

b. What is E[X] and Var[X]?

c. Find the probability that the number of runs required to produce an unacceptable lot is at least 3.

d. Suppose that 19 lots has been inspected and that all 19 are acceptable. What is theprobability that at least 21 acceptable lots are produced before the first unacceptable lot?

Explanation / Answer

random variable X flooows geometric distribution; with parameter p=0.05

b) E(X)=1/p =1/0.05=20

Var(X) =(1-p)/p2 =380

c)probability that the number of runs required to produce an unacceptable lot is at least 3=P(till 2 lot all lots are acceptable) =(1-0.05)2 =0.9025

d)as geomteric distribution is memoryless hence P(X>21|X>19)=P(X>2)=0.9025

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