# Thread: 2 questions: Finding the optimal value of the mean of a standard normal distribution.

1. ## 2 questions: Finding the optimal value of the mean of a standard normal distribution.

I have two questions involving the Standard Normal Distribution. The first one involves finding the optimal mean, the second one involves proving that the distance between two points a and b is the minimum length.

A machine makes shafts according to a Normal distribution with a standard deviation of .16 cm and a mean that can be calibrated by the operator. Shafts less than 55.05 cm are scrapped at a cost of $15.34 per shaft while those longer than 55.55 cm are reworked at a cost of$8.27 per shaft. The machine makes 10,000 shafts per week.

The machine is currently calibrated to have a mean of 55.250 cm. How should the machine be calibrated? What is the weekly cost for the optimal calibration?
I'm stuck on how to do this mathematically. I know I could guess and check, but that's not exactly mathematically rigorous, and it would take forever. I know that the mean has to be set higher than it is now, and I'm guessing it's somewhere around 55.37. That makes the lower value 2 standard deviations from the mean, making it far less likely that a shaft has to be scrapped completely while still not having to rework every shaft. I thought about using calculus to optimize the mean, but I can only think of one equation, and it's one that I can't figure out the derivative of.

My professor says that R allows me to construct a plot showing the scrap, rework, and total costs as a function of the calibrated value of the mean. I'm sure this would be helpful, but I'm unsure how to do this.

Generalize: For arbitrary 0<p<1, show that the method giving a and b in part c produces the minimum length interval.
I know that to minimize the distance between a and b for any value of p, they have to be equidistant from the mean. Using R, I can find values for a and b as follows:

a= qnorm(.5-p/2)
b=qnorm(.5+p/2)

I just don't know how to prove this, mathematically.

Thanks in advance for any help or hints you can give me.

2. ## Re: 2 questions: Finding the optimal value of the mean of a standard normal distribut

For the first one: You can express the expected cost in terms of the parameter (by Law of total expectation). Then minimize this expected cost by differentiation.

For the second one: I assume you working with a standard normal random variable satisfying , and you try to find the pair to minimize the distance

Note the condition is equivalent to . Here since is a constant, once either one of the is fixed, the other one is automatically uniquely determined.

So here e.g. you may write as a function of . Now differentiating both sides with respect to , we can obtain . On the other hand, due to the minimizing requirement (the first order condition). Make use of these facts you can prove your desired result. (Of course with the symmetric property as well)

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