ADDITIONAL PROBLEMS FOR LAST STAT 513 HW

Problem 1:

Consider the following life table, where Time is measured in years:

Time
Interval   n(t)   d(t)  w(t) 
--------   ----   ----  ----
[0,1)      300    90    10
[1,2)      200    40    5
[2,3)      155    50    5
[3,4)      100    25    10
[4,5)       65    10    12
[5,6)       43    10    3
[6,7)       30    5     5
[7,8)       20    7     3
[8,9)       10    6     1
[9,10)       3    1     2

(a) Give a point estimate and a 95% CI for the 3-year survival probability.


(b) Give a point estimate and a 95% CI for the 8-year survival probability.


Problem 2:

Consider the following lifetime data, with failure indicators:

# Survival or censoring times (in days):
time <- c(3,4,15,17,17,17,23,25,26,26,29,33,38,58,67,71,74,87,92,95,98,116,117,124,127,129, 151,155,156,162,167,179,186,204,204,209,209,214,225,227,228,228,232,239,240, 245,248,261,264,273,277,291,296,298,300,306,312,340,355,356,400,421,440,443,448,458,460, 486,506,517,526,571,613,634,641,666,704,754,757,774,780,794,871,887,989,1018,1673,1720,1794)

# Failure indicator:  1=death, 0=censored:
cens.ind <- c(1,1,1,0,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0, 1,1,0,0,0,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,0,0,1,1,0,1,1,1,0,1,1,0,1,0,1,1,1,1,0, 1,1,0,0,1,1,0,1,0,0,1,0,1,1,1,1,1,1,1)

(a) Give a point estimate and a 95% CI for the 1-year survival probability.

(b) Give a point estimate for the median survival time.


Problem 3:

The Rayleigh distribution has a pdf given on page 318 of the text (in Problem 6.34).  
Derive the survival function and the hazard function of the Rayleigh distribution, showing work.