(22 pts) Suppose that z follows a distribution with probability density function (p.d.f.) fz(z = x), x € R, i.e., P(a < z < b) = Sºfz(z = x)dx for any constants a < b.
a. (6 pts) Name three probability distributions, where the corresponding ran- dom variable takes continuous values (e.g., normal distributions). Write down the p.d.f. for each distribution, and its mean and variance (note: you may use the normal distribution in your answer). b. (2 pts) Write E[z] and var(z) as functions of fz(z = x), X E R. C. (2 pts) Write down the sample mean and the sample variance as functions of {z1, Z2, ..., Zn}. d. (4 pts) Write down the sample standard deviation of {Z1, Z2, ...,Zn}, and the (sample) standard error of the sample mean.
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Consider the following sample of observations on coating thickness for low-viscosity paint. ("Achieving a Target Value for a Manufacturing Process: A Case Study," J. of Quality Technology, 1992:22-26):
| 0.83 | 0.88 | 0.88 | 1.04 | 1.09 | 1.12 | 1.29 | 1.31 |
| 1.48 | 1.49 | 1.59 | 1.62 | 1.65 | 1.71 | 1.76 | 1.83 |
Assume that the distribution of coating thickness is normal (a normal probability plot strongly supports this assumption).
(a) Calculate a point estimate of the mean value of coating thickness, and state which estimator you used.
(b) Calculate a point estimate of the median of the coating thickness distribution, and state which estimator you used.
(c) Calculate a point estimate of the value that separates the largest 10% of all values in the thickness distribution from the remaining 90% and state which estimator you used. [Hint: Express what you are trying to estimate in terms of m and s]
(d) Estimate P(X<1.5), i.e., the proportion of all thickness values less than 1.5 [Hint: if you know the values of 
(e) What is the estimated standard error of the estimator that you used in part (b)?
