Question

A contractor estimates the probabilities for the number of days required to complete a certain type of project as follows:

Time (days) 1 2 3 4 5

Probability 0.04 0.21 0.34 0.31 0.10

(4) If the contractor's project cost is made up of two parts: a fixed cost of $100 million plus $10 million for each day taken to complete the project, find the variance of total project costs. (3 Marks) 

Answer 1

Solution:

First, we need to find the cost for each day, using formula: $f(x)=10x +100$ , where $x$ is the number of days. Calculating it for each day, we get the following distribution:

$\begin{Bmatrix} \xi \\ p(\xi) \end{Bmatrix} = \begin{Bmatrix} 110 & 120 & 130 & 140 & 150 \\ 0.04 & 0.21 & 0.34 & 0.31 & 0.1 \end{Bmatrix}$

Remembering the formula for variance:

$var(\xi) = E(\xi^2) - (E(\xi))^2$ ,

we also need to find distribution for $E(\xi^2)$ :

$\begin{Bmatrix} \xi^2 \\ p(\xi^2) \end{Bmatrix} = \begin{Bmatrix} 1210 & 1440 & 1690 & 1960 & 2250\\ 0.04 & 0.21 & 0.34 & 0.31 & 0.1 \end{Bmatrix}$

Next:

$E(\xi) = 0.04 * 110 + 0.21 * 120 + 0.34 * 130 + 0.31 * 140 + 0.1*150 = 132.2$

$E(\xi^2) = 0.04 * 12100 + 0.21 * 14400 + 0.34 * 16900 + 0.31 * 19600 + 0.1 * 22500 = 17580$

And we geat the answer:

$var(\xi) = E(\xi^2) - (E(\xi))^2 = 17580 - (132.2)^2 = 103.16$

Answer:

$var(\xi) =103.16$