Question

(5 marks) Consider an account of money that an insurance company has available to pay insurance claims, where the balance changes continuously over time according to the following rules: .The balance increases continuously at a constant rate of $2,000/day as premiums are collected The balance drops when a claim is paid out, where claim amounts, X, are Exponentially distributed with a mean of $1,000. The time to wait until the next claim, Y, is Exponentially distributed with a mean of 1 day. If the current balance in the account is $1,500, what is the probability that the next claim bankrupts the account? That is, what is the probability that after the next claim is paid, the balance will be below 0. Assume X and Y are independent. Hint: The balance after the next claim, in thousands, is the random variable, 1.5+2Y - X

Answer 1

We have to find the probability .

P(1.5 2Y-X< 0)

The PDFs are

(Xex> 0, fy(y) = e ¥ y> 0

The joint PDF is

e-y;X> 0, y 0 x,Y(X, y)

.

P(1.5 2Y -X< 0) P(1.5 +2Y < X) P(1.5 2Y X < 0) fx.y (x, y)dxdy J1.5+2y e x-Ydxdy 1.5+2y P(1.5 2Y X <0) e dxdy P(1.5 + 2Y - X < 0) = e y J1.5+2y e Ye 1.5-2ydy P(1.5 2Y X0) P(1.5 2Y X 0) e1.5 edy P(1.5+2Y X 0) e1.5/3 P(1.5 +2Y - X < 0) 0.0744