Solution Found!
In Exercise 5.41, we considered a quality control plan
Chapter 5, Problem 135E(choose chapter or problem)
Problem 135E
In Exercise 5.41, we considered a quality control plan that calls for randomly selecting three items from the daily production (assumed large) of a certain machine and observing the number of defectives. The proportion p of defectives produced by the machine varies from day to day and has a uniform distribution on the interval (0, 1). Find the
a expected number of defectives observed among the three sampled items.
b variance of the number of defectives among the three sampled.
Reference
A quality control plan calls for randomly selecting three items from the daily production (assumed large) of a certain machine and observing the number of defectives. However, the proportion p of defectives produced by the machine varies from day to day and is assumed to have a uniform distribution on the interval (0, 1). For a randomly chosen day, find the unconditional probability that exactly two defectives are observed in the sample.
Questions & Answers
QUESTION:
Problem 135E
In Exercise 5.41, we considered a quality control plan that calls for randomly selecting three items from the daily production (assumed large) of a certain machine and observing the number of defectives. The proportion p of defectives produced by the machine varies from day to day and has a uniform distribution on the interval (0, 1). Find the
a expected number of defectives observed among the three sampled items.
b variance of the number of defectives among the three sampled.
Reference
A quality control plan calls for randomly selecting three items from the daily production (assumed large) of a certain machine and observing the number of defectives. However, the proportion p of defectives produced by the machine varies from day to day and is assumed to have a uniform distribution on the interval (0, 1). For a randomly chosen day, find the unconditional probability that exactly two defectives are observed in the sample.
ANSWER:
Answer:
Step 1 of 2:
(a)
We considered a quality control plan that calls for randomly selecting three items from the daily production of a certain machine and observing the number of defectives.
The proportion of defectives produced by the machine varies from day to day and has a uniform distribution on the interval
We need to find the expected number of defectives observed among the three sampled items.
Let and
denote random variables. Then
…….(1)
Where on the right-hand side the inside expectation is with respect to the conditional distribution of given
and the outside expectation is with respect to the distribution of
.
Let be the number of defectives and
is the proportion of defectives produced by the varies uniformly from day to day.
Hence using equation (1) we can write,
……….(2)
For a given
has a binomial distribution.
We know the expected value of random variable , which follows a binomial distribution.
Hence we can write,
[ is given 3, number of sampled items ]
Thus equation (2) will become,
…….(3)
Since the proportion of defectives produced by the machine varies from day to day and has a uniform distribution on the interval
Hence the expected value of a random variable uniformly distributed on the interval (0, 1), then
Hence equation (3),
Hence the expected number of defectives observed among the three sampled items is