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

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    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)

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

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

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    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 \mu (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 Z satisfying \Pr\{a < Z < b\} = p, and you try to find the pair (a, b) to minimize the distance b - a

    Note the condition is equivalent to \Phi(b) - \Phi(a) = p. Here since p is a constant, once either one of the a, b is fixed, the other one is automatically uniquely determined.

    So here e.g. you may write b as a function of a. Now differentiating both sides with respect to a, we can obtain \frac {\partial b} {\partial a}. On the other hand, \frac {\partial} {\partial a} (b - a) = 0 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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