Question

4. The number of claims per week at an Suppose that Y has probability mass function insurance company is a random variable Y Pr(v),0,1,2. py(y) 0, otherwise. The moment generating function (mgf) of Y is given by my(t)-c(1-e2)- for values of t<2. You do not need to prove this. (a) Show that c1-2 (b) What is the probability that there are at least 2 claims in a given week? (c) Find E(Y)

Answer 1

(a)

,                                  (1)

P(y) = ce^-2y, (1) y = 0,1,2,... c is got by noting that the Total Probability = 1 Thus, we get: We note: e^-2 = 0.1353 So, we get: c[1 + 0.1353 + 0.1353^2 + ...] = 1 c(1 - 0.1353)^-1 = 1 So, c = 0.8647 So, Probability mass function is written as: P(y) = 0.8647 e^-2y P(y\geq2) = 1- [P(y = 0) + P(y = 1)] (2) P(y = 0) = 0.8647 e^-2 times 0 = 0.8647 p(y = 1) = 0.8647 e^-2 times 1 = 0.8647 times 0.1353 = 0.1170 So, P(y\geq2) = 1 - (0.8647 + 0.1170 = 1 - 0.9817 = 0.0183 So, Answer is: 0.0183

                        y = 0,1,2,...

c is got by noting that the Total Probability = 1

Thus, we get:

We note:

$e^{-2}=0.1353$

So, we get:

So,

c = 0.8647

(b)

So,

Probability mass function iswrittenas:

P(y $\geq$ 2) = 1- [P(y=0) + P(y=1)]                      (2)

$P(y=0)=0.8647e^{-2\times 0}=0.8647$

$P(y=1)=0.8647e^{-2\times 1}=0.8647\times 0.1353=0.1170$

So,

P(y $\geq$ 2) = 1 - (0.8647 + 0.1170 = 1 - 0.9817 = 0.0183

So,

Answer is:

0.0183