Stat 61 - Homework #6 (10/29/07, due 11/5/07)
1. Persistence of means
Start with 96 positive numbers whose mean, median, and 25%-trimmed mean are all exactly 60.
To this list, add four more values, all exactly equal to 60.
a. Show that the mean of all 100 numbers is 60, or show by example that the mean might not be 60.
b. Show that the median of all 100 numbers is 60, or show by example that the median might not be 60.
c. Show that the 25%-trimmed mean of all 100 numbers is 60, or show by example that the 25%-trimmed mean might not be 60.
2. Means computed from subsets
A certain set of M men has mean height A inches and median height C inches.
A certain set of W women has mean height B inches and median height D inches.
a. Give a formula for the mean height of all M+W of these people, or show by example that the combined mean height can’t be determined from the data given.
b. Give a formula for the median height of all M+W of these people, or show by example that the combined median height can’t be determined from the data given.
3. Increasing two means
One thing to say when a politician switches from your party to the other party is that he has increased the average intelligence of both parties.
Let’s talk about age instead of intelligence. Show by example that a professor who moves from a university’s Math Department to its Stat Department might well increase the average age of both departments.
4. Ranking the means
Let y1, y2, …, yn be positive numbers, not all the same. Let AM, GM, HM, and RMS denote their arithmetic mean, geometric mean, harmonic mean, and RMS mean respectively. Show that HM < GM < AM < RMS. (If this exercise seems to be taking too much time, assume that n = 3. Or even that n = 2.)
5. Estimating the parameter of an exponential distribution
A random variable Y has a gamma distribution with parameters a and b. We know that a = 1, but all we know about b is that b > 0.
The density function for Y is
for y ł 0.
Note that this is actually an exponential distribution with mean µ = 1/b. Still, we have chosen to write its density in terms of b.
Suppose now that we make four independent draws from this distribution. As random variables, we call them Y1, Y2, Y3, Y4. In fact, they are
y1 = 8.2, y2 = 9.1, y3 = 10.6, and y4 = 4.9.
a. Write the likelihood function, L(b). (Write it as a function of b, y1, y2, y3, y4, or just as a function of b using actual numbers for the observations.)
b. As a function of Y1, Y2, Y3, Y4, what is the maximum likelihood estimator for b?
c. What is the value of the estimate in this case?
d. (Maybe hard) Is this estimator unbiased? (That is, is it true that E() = ?)
e. Now let’s rewrite the density function as
for y ł 0.
Now µ is the unknown parameter (but still with µ > 0).
As a function of Y1, Y2, Y3, Y4, what is the maximum likelihood estimator for µ?
f. (Maybe not so hard) Is the estimator for µ unbiased?
g. Does the estimator in (e) clash with the estimator in (b) ?