.
A discrete random variable taking values 0, 1, . . . is said to have
a geometric distribution with parameter p if
P(N = k) = (1 − p) pk−1, k = 1, 2, . . . .
a. Suppose that X is exponential with parameter λ, and define a new random
variable N by
N = k if k − 1<>≤ k.
Show that N is geometric, and identify p.
b. Show that the mean EN of the geometric is 1/(1 − p), and calculate the
variance. [Hint: One way to do this is to complete the steps below:
For the variance, differentiate twice.]
c. The geometric arises as the distribution of the number of tosses up to and
including the first tail in a sequence of independent coin tosses in which
the probability of a head is p. Use this to calculate the distribution of N.
Can you derive EN using the waiting time analogy?