Question

Suppose Y is a normal random variable with mean 20 and standard deviation 4. A scientist observes the value for Y and uses it to construct a cube of dimensions |Y| * |Y| * |Y|.

(1) Find the expected surface area of the cube.
(2) Find the expected volume of the cube.
(3) Using computational software, approximate the probability that the volume is more than 5 times greater than the area.

Answer 1

Here Y is a normal random variable such that

1).

The surface area S= 6 so

Now Var(Y)= so

so

so the expected surface area is 2496 unit^2.

2)

so the expected volume is 8960 unit^3.

3) for question no. 3 I have computed manually using normal table. you can use computational software. I hope the answer will be almost the same. Thank you.