Question

3. It costs 30 cents per head per day to keep pigeons. Let N be the number of pigeons kept and suppose that N has the geometric distribution P(N = n) = 1/10* (9/10)^ n, (n = 0, 1, . . .).

(a) What is the expected daily cost

b) Now suppose that it costs an extra 5 cents per head per day for each pigeon in excess of 3 (if the number of pigeons exceeds 3, then each of 3 costs 30 cents per day and each of the others costs 35 cents per day). What now is the expected daily cost

Answer 1

Solution

Back-up Theory

If X has geometric distribution with parameter, p, its

pmf = P(X = k) = p(1 - p)^k, for k = 0, 1, 2, 3, ………..    ……………………………………. (1)

Mean: 1/p ……………………………………………………………………..………………… (2)

Now, to work out the solution,

Part (a)

Given N ~ Geometric(1/10),

E(N) = 10 [vide (2)]

So, expected number of pigeons kept = 10.

Given, ‘It costs 30 cents per head per day to keep pigeons.’,

the expected daily cost = 10 x 30 = 300 cents = $3 Answer 1

Part (b)

Here, out of 10 expected pigeons to be kept, 3 cost 30 cents each and the remaining 7 cost 35 cents each. So,

the expected daily cost = (3 x 30) + (7 x 35) = 335 cents = $3.35 Answer 2

DONE