1. Let X be a random variable and let EX = µ show that
(a) E(X – µ)2 = E(X2) – µ2.
(b) Var (aX + b) = a2 Var (X)
2. If X ~ N(µ, s2), show
X – µ
s ~ N(0, 1).
3. Consider the experiment of randomly drawing 100 observations from a uniform
distribution (0, 3). Compute the sample mean X¯ = 1
Xi. Repeat the experiments
500 times, plot the histogram of your estimated X¯. Is the mean of
X¯ = 1
X¯j close to EX? Does the histogram resemble the shape of a normal distribution?
4. There are 4,170 mutual funds reported in the Wall Street Journal on 01/06/1995.
By saying that performance relative to the market is random, we mean that each fund has a 50-50 chance of outperforming the market in any year and that performance is independent from year to year. Suppose Fund A indeed outperforms the market in all 10 years. Do you believe it is sheer luck or the firm manager is smarter than the market? Why or why not?
5. Suppose in a region the unemployment rate for male is.10 and the unemployment rate for female is .20. The unemployment rate of those with college degree is .05 and those without college degree is .25. Suppose half of the labor force is male
and half is female.
(a) What is the unemployment in the region?
(b) Can you derive the unemployment rate for male college graduate from the
information given. Why or why not?
(c) What additional information you will need to derive answers to (b) and (c).
Do you think there is discrimination against female college graduate based on additional assumptions that (i) half of the college graduates are male and half are female; (ii) the unemployment rate for female college graduate is 6%? Explain your answer.
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