Statistics Question

Question

A machine produces washers which must satisfy certain size and quality specifications. It produces a defective washer with probability p. Every hour a sample of ten washers is inspected and, if two or more washers are found to be defective, the machine is stopped and overhauled. What is the distribution of the time between stoppages? what is its expected value?

 

Answer 1

This is a geometric distribution. It is the discrete probability distribution that describes when the first success in an infinite sequence of independent and identically distributed Bernoulli trials occurs.

X ~ Geometric(p)

Probability mass function: P(Y = k) = p*q^k = p*(1−p)^k , where k = 0, 1, 2, … is the number of failures before the first success.

p = defective washer, where p is the success probability

q = working washer, where q is the failure probability

The expected value for the number of defective washers after testing 10 washers at random is 1/p.