The following problems are from FA&JR:
1. Let Y1;:::;Yn be i.i.d. U(0;#).
(a) Verify that Ymax=# is a pivot.
(b) Show that (Ymax; 1nYmax) is a (1 )100% con dence interval
for #.
2. Let Y1;:::;Yn be a random sample from the log-normal distribution
f ; (y) = 1p2 ye (lny )
2
2 2 , where both and are unknown. Find
100(1 )% con dence intervals for and .
(Hint: the distribution of X = lnY may be useful.)
3. Let t1;:::;tn exp(#). In Example ?? we constructed 95% con dence
intervals for #, = Et = 1=# and p = P(t > 5) = e 5# based on the
pivot 2#Pni=1ti 22n. Show that another pivot is #tmin exp(n). Use
it to construct 95% con dence intervals for #, , and p, and compare
them with those based on the mean of the observations.
4. A quality control laboratory tested n randomly chosen electronic de-
vices. The test lasted for a hours and at the end k devices out of n
were still operating. The operating time of a device is exponentially
distributed with an unknown mean . Based on the test’s results,
(a) Find the MLE of .
(b) Find a 100(1 )% con dence interval for .
(c) Find 100(1 )% con dence upper and lower bounds for .