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

(1) what is the probability that a randomly chosen project will take less than 3 days to complete?

(2) Find the expected time to complete a project.

(3) Find the standard deviation of time required to complete a project.

(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.

Answer 1

(1) The probability that a randomly chosen project will take less than 3 days to complete is equal

p₁+p₂ = 0.04+0.21 = 0.25

(2) The expected time to complete a project is equal

$E[X] = p_1+2p_2+3p_3+4p_4+5p_5$

$E[X] = 0.04 + 2\cdot 0.21 +3\cdot 0.34 +4\cdot0.31 + 5\cdot0.1 = 3.22$

(3) $E[X^2] = p_1+2^2p_2+3^2p_3+4^2p_4+5^2p_5$

$E[X^2] = 0.04 + 4\cdot 0.21 +9\cdot 0.34 +16\cdot0.31 + 25\cdot0.1 = 11.4$

$Var[X] = E[X^2] - E[X]^2$

$Var[X] = 11.4 - 3.22^2 = 1.0316$

The standard deviation of time required to complete a project is equal

$\sigma_X=\sqrt{Var[X]} = \sqrt{1.0316} = 1.0157$