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Suppose that the quality of a widget is distributed according to a score, given by a normal distribution with mean 1 and variance σ^2. A fraction, π of all widgets are defective. The cost of having an inspector to determine if the widget is faulty, with certainty, is c. If the product is not faulty, the company incurs the cost c. If the product is faulty, the company incurs a "negative cost" b, which is (r - c), where r is the "cost avoided" by not shipping the faulty widget.

For what values of the score should the company have a product inspected? The standard normal CDF Φ(x) should be used.

What I've come up with so far:

Given a specific value for c and b, there is a break even point for the accuracy of sampling on the interval [0, 1]. Call this point m.

As long as (m * b) - (1 - m) * c > 0, then the company is making money.

So, if π > (1 - m), I would sample items with a score 1 + Φ(1 -m) Otherwise, I would sample items with a score 1 + Φ(π)

That's my answer so far, and I don't see how Bayesian inference comes into play (which was the topic of this week's lecture in this class). Am I supposed to modify the value of π with additional observations? If so, how do I do this? What prior do I use?

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