5.2 Point Estimates 59Now we will describe another important discrete distribution the
Poisson distribution. In clinical trials, we often consider the time from
entrance in the study to the occurrence of a particular event as an end-
point. We will cover this in more depth when we reach the survival
analysis topic. One of the simplest parametric models of time to an
event is the exponential distribution. This distribution involves a single
parameter λ called the rate parameter. It is a good model for some time
to failure data, such as light bulbs. For the exponential distribution, the
probability that the time to the fi rst event is less than t is 1 − exp( − λ t )
for 0 ≤ t < ∞. The Poisson distribution is related to the exponential
distribution in the following way: It counts the number of events that
occur in an interval of time of a specifi ed length t (say t = 1).
We have the following relationship: Let N be the number of events
in the interval [0, 1] when events occur according to an exponential
distribution with parameter λ. Let the exponential random variable be
T. Then P [ N ≥ k ] = P [ T ≤ 1] k = [1 − exp( − λ )] k. This relates the Poisson
to the exponential mathematically. This says that there will be at least k
events in [0, 1] as long as the fi rst k events are all less than 1. So:
PN k[ <=−− − = ≤−] 11 [ exp( λ)]k PN k[ 1 ].
For k ≥ 1, using the binomial expansion for [1 − exp( − λ )] k , we can
derive the cumulative Poisson distribution.
The following fi gure shows an example of a binomial distribution
with n = 12 and p = 1/3. In the fi gure, π is used to represent the param-
eter p. The Poisson distribution is also given for λ = 0.87 and t = 1.
Note that the binomial random variable can take on integer values from
0 to 12 in this case, but the Poisson can be any integer greater or equal
to zero (though the probability that N > 5 is very small. Also note that
the probability that the number of successes is 11 is very small, and for
12, it is even smaller, while the probability of 0 or 1 success is much
larger than for 11 or 12. This shows that this binomial is skewed to the
right. This Poisson is also skewed right to an even larger extent (Figs.
5.1 and 5.2 ).
5.2 POINT ESTIMATES
In Chapter 3 , we learned about summary statistics. We have discussed
population parameters and their sample analogs for measures of central