Discrete Mathematical Structures by Pearson

Combinatorics 5-49

5.14.2 Another Approach to Poisson Distribution

Let the random variable X denote the number of telephone calls during time t at a telephone

switch board. We assume that (1) the calls are independent and (2) the probability of a call in

time dt is l dt where l is a real number.

Let P

x

(t) be the probability of x calls in time t, then P

x

(t + dt) will denote the probability of x

calls in time t + dt. Now two mutually exclusive cases will arise.

1. x calls in time t and no call in time dt

2. (x - 1) calls in time t and one call in time dt

(The possibility of more than one call in time dt is neglected)

∴ P

x

(t + dt) = P

x

(t)(1-λdt) ...