Answered

At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Get immediate and reliable answers to your questions from a community of experienced professionals on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.

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

Sagot :

MrRoyal

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]

Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.