Problem 22 An insurance company will cover losses incurred from tornadocs in a s

Question

Problem 22 An insurance company will cover losses incurred from tornadocs in a singloc calendar year. However, the insurer will only cover losses for a maximum of three separate tornadoes during this time frame. Let X be the number of tornadoes that result in at least 50 million in losses, and let Y be the total number of tornadoes. The joint probability function for X and Y is -J c(x + 2y)· = 0. 1. 2. 3. ? 0. 1. 2. 3. x-y otherwise where c is a constant. Calculate the expected number of tornadoes that re- sult in fewer than 50 million in losses (A) 0.19 (B) 0.28 (C) 0.76 (D) 1.00 (E) 1.10

Explanation / Answer

First the value of x is computed by using the fact that the sum of all probabilities is always equal to 1.

P( 0, 1) + P(1, 1) + P(0, 2) + P(1, 2) + P(2, 2) + P(0, 3) + P(1, 3) + P(2, 3) + P(3, 3) = 1

2c + 3c + 4c + 5c + 6c + 6c + 7c + 8c + 9c = 1

50c = 1

c = 1/50 = 0.02

Therefore, we get here:

P(X = 0) = P(0, 1) + P(0, 2) + P(0, 3) = 12c = 0.24
P(X = 1) = P(1, 1) + P(1, 2) + P(1, 3) = 15c = 0.30
P(X = 2) = P(2, 2) + P(2, 3) = 14c = 0.28
P(X = 3) = P(3, 3) = 9c = 0.18

Therefore E(X) = 0.24*0 + 0.3*1 + 0.28*2 + 0.18*3 = 1.4

P(Y = 0) = 0
P(Y = 1) = P(0, 1) + P(1, 1) = 5c = 0.10
P(Y = 2) = P(0, 2) + P(1, 2) + P(2, 2) = 15c = 0.30
P(Y = 3) = P(0, 3) + P(1, 3) + P(2, 3) + P(3, 3) = 30c = 0.60

Therefore E(Y) = 1*0.1 + 2*0.3 + 3*0.6 = 2.5

Therefore E(Y) - E(X) = 2.5 - 1.4 = 1.1 which is the required expected value.

Therefore 1.1 is the expected value here.

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