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Answer:
The variance of total loss is 8000000
Step-by-step explanation:
Let
[tex]X \to[/tex] Number of hurricane
Poisson [tex]E(X) = 4[/tex]
[tex]Y \to[/tex] Loss in each hurricane
Exponential [tex]E(Y) = 1000[/tex]
[tex]T \to[/tex] Total Loss
Required
The variance of the total loss
This is calculated as:
[tex]Var(T) = Var(E(T|X)) + E(Var(T|X))[/tex]
Where:
[tex]E(T|X) \to[/tex] Expected total loss given X hurricanes
And it is calculated as:
[tex]E(T|X) = E(Y) *N[/tex] --- Expected Loss in each hurricane * number of loss
[tex]Var(T|X) \to[/tex] Variance of total loss given X hurricanes
And it is calculated as:
[tex]Var(T|X) = Var(Y) * N[/tex] ---- --- Variance of loss in each hurricane * number of loss
So, we have:
[tex]Var(T) = Var(E(T|X)) + E(Var(T|X))[/tex]
[tex]Var(T) = Var(E(Y) * N) + E(Var(Y) * N)[/tex]
For exponential distribution;
[tex]Var(Y) = E(Y)^2[/tex]
So, we have:
[tex]Var(T) = Var(E(Y) * X) + E(E(Y)^2 * X)[/tex]
Substitute values
[tex]Var(T) = Var(1000 * X) + E(1000^2 * X)[/tex]
Simplify:
[tex]Var(T) = Var(1000 * X) + 1000^2E(X)[/tex]
Using variance formula, we have:
[tex]Var(T) = 1000^2Var(X) + 1000^2E(X)[/tex]
For poission distribution:
[tex]Var(X) = E(X)[/tex]
So, we have:
[tex]Var(X) = E(X) = 4[/tex]
The expression becomes:
[tex]Var(T) = 1000^2*4 + 1000^2*4[/tex]
[tex]Var(T) = 1000000*4 + 1000000*4[/tex]
[tex]Var(T) = 4000000 + 4000000[/tex]
[tex]Var(T) = 8000000[/tex]