Question

Suppose that the number of printing mistakes on each page of a 200-page Mathematics book is independent of that on other pages. and it follows a Poisson distribution with mean 0.2.

(a) Find the probability that there is no printing mistake on page 23.

(b) Let page N be the first page which contains printing mistakes.

Find (i) the probability that N is less than or equal to 3,

(ii) the mean and variance of N.

(c) Let M be the number of pages which contain printing mistakes.Find the mean and variance of M.

(d) Suppose there is another 200-page Statistics book and there are 40 printing mistakes randomly and independently scattered through it.

Let Y be the number of printing mistakes on page 23.

(i) Which of the distributions - Bernoulli, binomial, geometric, Poisson, does Y follow?

(ii) Find the probability that there is no printing mistake on page 23.

Answer 1

(a)

Let X be the number of printing mistake on a random page. Then X ~ Poisson( = 0.2)

Probability that there is no printing mistake on page 23 = P(X = 0)

= exp(-0.2) * 0.2⁰ / 0!

= 0.8187

(b)

Probability of at least one mistake on a page = 1 - Probability that there is no printing mistake on page

= 1 - 0.8187 = 0.1813

Let page N be the first page which contains printing mistakes. Then N will follow Geometric distribution with the parameter p = 0.1813

(i) Probability that N is less than or equal to 3 = P(N 3)

= 1 - (1 - 0.1813)³ (Using CDF of Geometric distribution)

= 0.4513

(ii)

Mean of N = 1/p = 1/0.1813 = 5.5157

Variance of N = (1 - p)/p² = (1 - 0.1813) / 0.1813² = 24.9075

(c)

Let M be the number of pages which contain printing mistakes. Then M will follow Binomial distribution with the parameters n = 200 and p = 0.1813

Mean = np = 200 * 0.1813 = 36.26 pages

Variance of M = np(1-p) = 200 * 0.1813 * (1 - 0.1813) = 29.686

(d)

Average number of printing mistakes per page = 40/200 = 0.2

Since the outcomes of number of printing mistakes on page can have multiple values, Y will follow Poisson distribution.

(ii)

Probability that there is no printing mistake on page 23 = P(X = 0)

= exp(-0.2) * 0.2⁰ / 0!

= 0.8187