Let X have an exponential distribution.
(a) For 
and 
, show that
Hence, the exponential distribution has the memory less property. Recall from Exercise 3.1.9 that the discrete geometric distribution has a similar property.
(b) Let 
be the cdf of a continuous random variable Y . Assume that 
Exercise 3.1.9
If 
is the unique mode of a distribution that is 
, show that
This substantiates the comments made in Part (b) of Exercise 3.1.8.
Exercise 3.1.8
For this exercise, the reader must have access to a statistical package that obtains the binomial distribution. Hints are given for R code, but other packages can be used too.
(a) Obtain the plot of the pmf for the b(15, 0.2) distribution. Using R, the following commands return the plot:
(b) Repeat part (a) for the binomial distributions with n = 15 and with p = 0.10, 0.20,…, 0.90. Comment on the shapes of the pmf’s as p increases. Use the following R segment:
