Mean and Variance of Linear Functions of a Random Variable

Let X be a discrete random variable with probability function P(x), and let g(x) be some function of X. Then the expected value, E[g(X)], of that function is defined as

Summary of Properties for Linear Functions of a Random Variable

Let X be a random variable with mean and variance , and let a and b be any constant fixed numbers. Define the random variable Y as (a+bX)  Then, the mean and variance of Y are

Example: A contractor is interested in the total cost of a project on which he intends to bid. He estimates that materials will cost $25,000 and that his labor will be $900 per day. If the project takes X days to complete, the total labor cost will be 900X dollars., and the total cost of the project (in dollars) will be

C=25000+900X                       
The contractor forms subjective probabilities of likely completion times for the project as follows:
Table: Probability distribution for completion times

Completion time X (days)	10	11	12	13	14
Probability	0.1	0.3	0.3	0.2	0.1

a.    Find the mean and variance for completion time X.
b.    Find the mean, variance, and standard deviation for total cost C.

Solution:

a.     The mean and variance for completion time X can be found as

b.    The mean, variance and standard deviation of total cost, C, are obtained as follows:
The mean is