Homework 7 for BST 631: Statistical Theory I – Problems, 10/05/2006
1
Due Time: 5:00PM Thursday, on 10/12/2006.
Problem 1 (20 points). Book problem 3.24 (b) and (d).
Problem 2 (10 points). book problem 3.26(b).
Problem 3 (12 points). Book problem 3.33(b) and (c).
Problem 4 (15 points). Book problem 3.39.
Problem 5 (8 points). Book problem 3.40.
Problem 6 (10 points). Book problem 3.45(a) and (b).
Problem 7 (25 points). (Qualify Exam 2005 July 22 Problem 3)
The time to death (in days) after an autologous bone marrow transplant follows a log normal
distribution with 3.177
µ
= and 22.084
σ
=. Hint: For the log-normal distribution, we can write
((logt ) / )
()ft t
φ
µσ
σ
−
= and () ( ) 1 [(ln )/ ]St PT t t
µ
σ
=>=−Φ− , where ()
φ
⋅ and ()Φ⋅ represent
the pdf and cdf, respectively, of a standard normal random variable.
(1) Find the mean and median times to death.
(2) Find the probability that an individual survives 10, 50 and 100 days following a transplant.
(3) In survival analysis, the hazard function is used to measure the risk that an individual
experiences the event of interest. The higher the hazard function at a given time point, the greater
the risk. Taking advantage of the fact that the hazard rate may be written as ( ) ( ) / ( )
ht f t St=,
plot the hazard rate of the time to death at t = 0.1, t =0.35, t = 2, t = 10, t = 50, and t = 100 days.
What does the shape of this function tell you about the hazard of death over time?
(4) Derive the general formula for the variance of a random variable T that has a log normal
distribution with parameters
µ
and 2
σ
. (Hint: You may find it useful to use the mgf of a normal
random variable.)