The Arithmetic of Operations on Statistics and Random Variables

When it comes to arithmetic, random variables are not much different than deterministic variables. We can add, subtract, multiply, and divide random variables. For instance, we can define a random variable Z = X + Y, or W = X². We can transform a random variable into a deterministic variable by calculating its expected value. However, many functional and logical operations on random variables depend on whether or not the variables are mutually exclusive or independent. As examples, the functional operation of expected value does not depend on independence, but the functional operation of variance does.

Similarly, there are operations on statistics that both inherit their properties from deterministic variables and acquire certain properties from the nature of randomness. For instance, the variance of a sum is the sum of variances if the random variables are independent, but the standard deviation of the sum is not the sum of the standard deviations.

Table 2-4 provides a summary of the most important operations for project managers.

Table 2-4: Operations on Random Variables and Statistics
Item	All Arithmetic Operations	All Functional Operations with Random Variables as Arguments	Limiting Conditions
Random variables	Yes	Yes
Probability density functions	Yes	Yes
Cumulative probability density functions	Yes	Yes	If a random variable is dependent upon another, the functional expression is usually affected.
Expected value, or mean, or sample average, or arithmetic average	Yes	Yes
Variance	Yes	Yes	If the random variables are not independent, then a covariance must be computed.
Standard deviation	Cannot add or subtract	Yes	To add or subtract standard deviations, first compute the sum of the variances and then take the square root.
Median	No	No	Median is calculated on the population or sample population of the combined random variables.
Mode or most likely	No	No	Most likely is taken from the population statistics of the combined random variables.
Optimistic and pessimistic random variables	Yes	Yes	None