Question

The number of hurricanes that will hit a certain house in the next ten years is Poisson distributed with mean 4. Each hurricane results in a loss that is exponentially distributed with mean 1000. Losses are mutually independent and independent of the number of hurricanes. Calculate

Answer 1

Answer:

The variance of total loss is 8000000

Step-by-step explanation:

Let

$X \to$ Number of hurricane

Poisson $E(X) = 4$

$Y \to$ Loss in each hurricane

Exponential $E(Y) = 1000$

$T \to$ Total Loss

Required

The variance of the total loss

This is calculated as:

$Var(T) = Var(E(T|X)) + E(Var(T|X))$

Where:

$E(T|X) \to$ Expected total loss given X hurricanes

And it is calculated as:

$E(T|X) = E(Y) *N$ --- Expected Loss in each hurricane * number of loss

$Var(T|X) \to$ Variance of total loss given X hurricanes

And it is calculated as:

$Var(T|X) = Var(Y) * N$ ---- --- Variance of loss in each hurricane * number of loss

So, we have:

$Var(T) = Var(E(T|X)) + E(Var(T|X))$

$Var(T) = Var(E(Y) * N) + E(Var(Y) * N)$

For exponential distribution;

$Var(Y) = E(Y)^2$

So, we have:

$Var(T) = Var(E(Y) * X) + E(E(Y)^2 * X)$

Substitute values

$Var(T) = Var(1000 * X) + E(1000^2 * X)$

Simplify:

$Var(T) = Var(1000 * X) + 1000^2E(X)$

Using variance formula, we have:

$Var(T) = 1000^2Var(X) + 1000^2E(X)$

For poission distribution:

$Var(X) = E(X)$

So, we have:

$Var(X) = E(X) = 4$

The expression becomes:

$Var(T) = 1000^2*4 + 1000^2*4$

$Var(T) = 1000000*4 + 1000000*4$

$Var(T) = 4000000 + 4000000$

$Var(T) = 8000000$