Question

An individual who has automobile insurance from a certain company is randomly selected. Let Y be the number of moving violations for which the individual was cited during the last three years. The p.m.f. of Y is: P(Y = 0) = 0.6, P(Y = 1) = 0.25, P(Y = 2) = 0.1 and P(Y = 3) = 0.05.

(a) Compute E(Y ), E(Y 2 ) and E(2Y 2 − 4Y + 5).

(b) Suppose an individual with Y violations incurs a surcharge of $100Y 2 . Calculate the expected amount of the surcharge.

Answer 1

a) The expected values here are computed as:
E(Y) = 0*0.6 + 1*0.25 + 2*0.1 + 3*0.05 = 0.6
therefore 0.6 is the expected value here.

E(Y²) = 1²*0.25 + 2²*0.1 + 3²*0.05 = 1.1
Therefore 1.1 is the required expected value here.

E(2Y² - 4Y + 5) = 2E(Y²) - 4E(Y) + 5 = 2*1.1 - 4*0.6 + 5 = 4.8
Therefore 4.8 is the required expected value here.

b) The surcharged here is given as:
S = 100Y²

Therefore the expected amount of the surcharge here is computed as:

E(S) = 100E(Y²) = 100*1.1 = 110

Therefore 110 is the expected surcharge here.