(a) Show that if N is a Poisson random variable with mean \lambda, then its probability
generating function is given by
$P_N(z) = e^{\lambda(z-1)}$
(b) Cyber risk for a firm is based on its liability for a data breach involving sensitive
customer information, such as Social Security numbers, credit card numbers, ac-
count numbers, driver's license numbers and health records. A company models
its cyber risk by believing that the distribution of cyber events is a function of
its technical support staff size that has increased over time. Thus, it models the
number of cyber events (at each quarter) as a Poisson distribution with expected
number of events as:
Quarter
1 2 3 4 5 6 7 8 9 10 11 12
Expected Number 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.3 0.3 0.4 0.5
Assuming the numbers of cyber events in different calendar quarters are mutually
independent, determine (with proof) the probability mass function of the total
number of cyber events over the three year period (12 quarters). Hence calculate
the pmf and the cumulative probability distribution function for k = 0, 1, 2, ..., 12
cyber events (over the three-year period).
