Question

The intensity of a hurricane is a random variable that is uniformly distributed on the interval [0, 3]. The damage from a hurricane with a given intensity y is exponentially distributed with a mean equal to y. Calculate the variance of the damage from a random hurricane.

Answer 1

3.75

Explanation:

We have that x is the random variable denoting damage and y is the variable denoting intensity:

they tell us that the interval is [0.3], the variance in this case is the same:

var (x) = var * [E (x | y)] + E * [var (x | y)]

from here we have to:

var (x) = var (y) + E (y ^ 2)

we know that var (y) = E (y ^ 2) - E ^ 2 (y), we replace

var (x) = E (y ^ 2) - E ^ 2 (y) + E (y ^ 2)

var (x) = 2 * E (y ^ 2) - E ^ 2 (y)

We have that E (y ^ 2) = integral from 0 to 3, from 1/3 y ^ 2 * dy

we solve and we are left with:

1/9 and ^ 3, from 0 to 3

(1/9) * (3 ^ 3) - (1/9) * (0 ^ 3) = 3

We also know that E (y) = 3/2

replacing we have:

var (x) = 3 * 2 - (3/2) ^ 2

var (x) = 3.75

Therefore the variance is 3.75