Page 76 - Probability, Random Variables and Random Processes

68 RANDOM VARIABLES [CHAP 2

2.40. Show that the Poisson distribution can be used as a convenient approximation to the binomial
distribution for large n and small p.
From Eq. (2.36), the pmf of the binomial r.v. with parameters (n, p) is

n(n - 1Xn - 2) . . (n - k + 1) pk(l - p)"-k
p = ()l - p). - =
k !
Multiplying and dividing the right-hand side by nk, we have
- i)(l - ) . . - 7)
(np)k(l- :r-k
( I - p). - = k !

If we let n + oo in such a way that np = 1 remains constant, then

where we used the fact that

Hence, in the limit as n -, oo with np = 1 (and as p = Iln + O),

Thus, in the case of large n and small p,

which indicates that the binomial distribution can be approximated by the Poisson distribution.

2.41. A noisy transmission channel has a per-digit error probability p = 0.01.
(a) Calculate the probability of more than one error in 10 received digits.
(b) Repeat (a), using the Poisson approximation Eq. (2.100).
(a) It is clear that the number of errors in 10 received digits is a binomial r.v. X with parameters (n, p) =
(10,0.01). Then, using Eq. (2.36), we obtain

(b) Using Eq. (2.100) with 1 = np = 1q0.01) = 0.1, we have

2.42. The number of telephone calls arriving at a switchboard during any 10-minute period is known
to be a Poisson r.v. X with A= 2.