Question

The pessimistic time and optimistic time of completion of an activity are given as 10 days and 4 days respectively. Then the variance of the activity is

• A. 14
• B. 6
• C. 2
• D. 1

Hint:

Be careful not to confuse standard deviation with variance! - $\text{Standard Deviation} = \frac{t_p - t_o}{6}$ - $\text{Variance} = \left(\frac{t_p - t_o}{6}\right)^2$ Always make sure to square your result when a question asks for the variance.

Answer 1

The Correct Option is D

Concept: Within the Project Evaluation and Review Technique (PERT) framework, the completion duration of any specific activity is treated as a random variable modeled via a Beta probability distribution. To describe this distribution, three time estimates are collected: 1. Optimistic time ($t_o$ or $a$): Minimum possible timeframe if everything goes perfectly. 2. Most likely time ($t_m$ or $m$): The modal or standard duration expected under typical circumstances. 3. Pessimistic time ($t_p$ or $b$): Maximum duration required if significant obstacles are encountered. In PERT theory, the standard deviation ($\sigma$) of an activity's duration is estimated by dividing the total practical span of the distribution by 6 standard deviations: \[ \sigma = \frac{t_p - t_o}{6} \] The statistical variance ($\sigma^2$) is the square of this standard deviation: \[ \text{Variance} = \sigma^2 = \left(\frac{t_p - t_o}{6}\right)^2 \]

Step 1: Isolate variables from the text description.
The problem provides the following parameters: * Pessimistic completion time, $t_p = 10\text{ days}$ * Optimistic completion time, $t_o = 4\text{ days}$

Step 2: Calculate the standard deviation ($\sigma$).
Substitute our parameters directly into the linear PERT approximation equation: \[ \sigma = \frac{10 - 4}{6} \] \[ \sigma = \frac{6}{6} = 1\text{ day} \]

Step 3: Calculate the activity variance ($\sigma^2$).
Square the standard deviation value computed in the previous step: \[ \text{Variance} = \sigma^2 = (1)^2 = 1 \] The computed variance is exactly $1$, which corresponds to Option (4).