Standard Probability Distributions - Advanced Probability Theory for Biomedical Engineers - page 18

hence,

E k −1 i )

k

( x − = λ . (5.53)

i = 0

2

E ( x 2 )

x 2 2

In particular, E ( x ) .

While it is quite easy to calculate the value of the Poisson PMF for a particular number
of successes, hand computation of the CDF is quite tedious. Therefore, the Poisson CDF is
tabulated in Tables A.4-A.7 of the Appendix for selected values of

= λ , E ( x ( x − 1)) = λ = − λ , so that σ = λ + λ − λ = λ

ranging from 0.1 to 18.
From the Poisson CDF table, we note that the value of the Poisson PMF increases as the number
of successes k increases from zero to the mean, and then decreases in value as k increases from
the mean. Additionally, note that the table is written with a finite number of entries for each
value of

λ

because the PMF values are written with six decimal place accuracy, even though an
infinite number of Poisson successes are theoretically possible.

λ

Example 5.4.1. On the average, Professor Rensselaer grades 10 problems per day. What is the
probability that on a given day (a) 8 problems are graded, (b) 8-10 problems are graded, and (c) at
least 15 problems are graded?

Solution. With x a Poisson random variable, we consult the Poisson CDF table with λ = 10,

and find

a) p x (8) = F x (8) − F x (7) = 0 . 3328 − 0 . 2202 = 0 . 1126,

b) P (8 ≤ x ≤ 10) = F x (10) − F x (7) = 0 . 5830 − 0 . 2202 = 0 . 3628,

c) P ( x ≥ 15) = 1 − F x (14) = 1 − 0 . 9165 = 0 . 0835.

5.4.1 Interarrival Times
In many instances, the length of time between successes, known as an interarrival time, of a
Poisson random variable is more important than the actual number of successes. For example,
in evaluating the reliability of a medical device, the time to failure is far more significant to the
biomedical engineer than the fact that the device failed. Indeed, the subject of reliability theory
is so important that entire textbooks are devoted to the topic. Here, however, we will briefly
examine the subject of interarrival times from the basis of the Poisson PMF.
Let RV t r denote the length of the time interval from zero to the r th success. Then

p (

τ − < ≤ τ ) =

p ( r

− 1 ,τ −

h ) p (1

,

h ) h t r

=

p ( r

− 1 ,τ −

h )

λ +

o ( h )

h