Solved: "You are given: (i) The annual number of..."

kiki195ms

Answered question

2021-11-26

You are given:
(i) The annual number of claims on a given policy has the geometric distribution.
(ii) One-third of the policies have two claims

(β = 2)

, abd the remaining two-thirds have five claims

(β = 5)

Answer & Explanation

oces3y

Beginner2021-11-27Added 21 answers

Step 1
As per given:
The annual number of claims on a given policy has the geometric distribution.
And one third of the policies have two claims, and remaining two thirds have five claims.
For the geometric distributions:

P (X_{1} = 2 β) = \frac{β^{2}}{{(1 + β)}^{3}}

The expected value is

β

.
Step 2
Solving it:

P (β = 2 ∣ X_{1} = 2) = \frac{P (X_{1} = 2 ∣ β = 2) P (β = 2)}{P (X_{1} = 2 ∣ β = 2) P (β = 2) + P (X_{1} = 2 ∣ β = 5) P (β = 5)}

P (β = 2 ∣ X_{1} = 2) = \frac{\frac{4}{27} \times \frac{1}{3}}{\frac{4}{27} \times \frac{1}{3} + \frac{25}{216} \times \frac{2}{3}}

P (β = 2 ∣ X_{1} = 2) = 0.39024

Hence the expected value is:

β = (0.39024) (2) + (0.60976) (5)

β = 3.83

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