Question

a) The solar installation generates electric power from sunlight and incurs operating costs for cleaning
the solar modules (sometimes called solar panels) and replacing solar modules that have failed. The
annual revenue from the electric power is variable due to variable cloudiness and solar module failure
and has a mean of $2.78m and a standard deviation of $0.32m. The annual operating costs have a mean
of $0.51 and a standard deviation of $0.12m.

calculate the mean and standard deviation of the annual profit ​

Answer 1

Answer:

The mean of the annual profit is $2.27 m.

The standard deviation of the annual profit is $0.3418 m.

Explanation:

Let X represent the annual operating costs for cleaning  the solar modules and replacing solar modules that have failed.

And let Y represent the annual revenue from the electric power.

It is provided that:

$E(X)=\ 0.51\ m\\E(Y)=\ 2.78\ m\\SD(X)=\ 0.12\ m\\SD(Y)=\ 0.32\ m$

It is known that:

Annual Profit (Z) ​= Annual Revenue (Y) - Annual Operating Costs (X)

Compute the mean of the annual profit ​as follows:

$E(Z)=E(Y-X)$

        $=E(Y)-E(X)\\=2.78-0.51\\=2.27$

The mean of the annual profit is $2.27 m.

Compute the standard deviation of the annual profit ​as follows:

$SD(Z)=\sqrt{V(Y-X)}$

        $=\sqrt{V(Y)+V(X)-2Cov(X,Y)}\\=\sqrt{V(Y)+V(X)}\\=\sqrt{(0.32)^{2}+(0.12)^{2}}\\=0.3418$

The standard deviation of the annual profit is $0.3418 m.