Question

Acme typically produces three engines per day. Some of the engines pass the quality control inspection on the first try and are ready to be shipped; others need to be retooled. The probability of an engine needing further work is 0.05. If the engine is ready to be shipped, the company makes a profit of $10,000. If it needs to be retooled, it ultimately costs the company $2000. 

(a) Find the pmf of the number of engines that are ready to be shipped. 

(b) Find the pmf of the company's daily profit for the plant. 

(c) Find the expected profit.

 

 

Answer 1

Acme Industries typically produces three electric power generators a day; some pass the company’s quality-control inspection on their first try and are ready to be shipped; others need to be retooled. The probability of a generator needing further work is 0.05. If a generator is ready to ship, the firm earns a profit of $10,000. If it needs to be retooled, it ultimately costs the firm $2,000. Let X be the random variable quantifying the company’s daily profit.

 

The underlying sample space here is a set of n = 3 independent trials, where 

p = P(Generator passes inspection) = 0.95. 

 

If the random variable X is to measure the company’s daily profit, then

 

For instance, X(s, f, s) = 2($10,000) – 1($2,000) = $18,000. Moreover, the random variable X equals $18,000 whenever the day’s output consists of two successes and one failure. That is, X(s, f, s) = X(s, s, f) = X(f, s,s).

 

It follows that:

 

Table below shows pX(k) for the four possible values of k ($30,000, $18,000, $6,000, and −$6,000).

No. Defectives	k = profit	pX(k)
0	$30,000	0.857375
1	$18,000	0.135375
2	$6,000	0.007125
3	−$6,000	0.000125

 

The expected value of X is denoted by E(X) and is given by:

 

Hence,