Random Variables

A random variable is a variable that takes on numerical values determined by the outcome of a random experiment.

It is important to distinguish between a random variable and the possible values that it can take. Notationally, this is done by using capital letters, such as X to denote the random variable and the corresponding lowercase letter, x, to denote a possible value. For example, prior to the results being observed in the throw of a die, we can use the random variable X to denote the outcome. This random variable can take the specific values x = 1, x = 2, ….., x = 6, each with probability P(X = 1)=……………P(X = 6) = 1/6

Discrete Random Variable

A random variable is a discrete random variable if it can take on no more than a countable number of values.

Some examples of discrete random variables are:

1.     The number of defective items in a sample of 20 items from a large shipment.

2.     The number customers arriving at a checkpoint counter in an hour.

3.     The number of errors detected in a corporation’s accounts.

4.     The number of claims on a medical insurance policy in a particular year.

Continuous Random Variable

A random variable is a continuous random variable if it can take any value in an interval.

For, continuous random variables we cannot assign probabilities to specific values. For example, the probability that today’s high temperature will be precisely 77.236 degree Fahrenheit is 0. The temperature will certainly not be precisely that figure. However, probabilities may be determined for ranges, so that one could attach a probability to the event “Today’s high temperature will be between 75 and 80 degree.”

Probability Distribution Function

The probability distribution function, P(x), of a discrete random variable X expresses the probability that X takes the value x, as a function of x.
That is,
                 P(x) = P(X = x),  for all values of x

Example: Rolling a Die (Probability Function Graph)
Graph the probability distribution function for the roll of a single six-sided balanced die.

Solution: Let the random variable X denote the number resulting from a single roll of a six-sided balanced die. Since

The function takes the value 0 for all other values of x, which cannot occur. The probability distribution function is graphed in the above figure,

Required Properties of Probability Distribution

Let X be a discrete random variable with probability distribution function P(x). Then

Cumulative Probability Function

The cumulative probability function, F(Xo), for a random variable X, expresses the probability that X does not exceed the value Xo, as a function of Xo. That is, where the function is evaluated at all values of Xo.
Example: Stetson Motors, Inc., is a car dealer in a small Midwestern town. Based on an analysis of its sales history, the managers know that on any single day the number of Vertigo A cars sold can vary from 0 to 5. How can the probability distribution function shown in the following table be used for inventory planning.

x	P(x)	F(x)
0	0.15	0.15
1	0.30	0.45
2	0.20	0.65
3	0.20	0.85
4	0.10	0.95
5	0.10	0.95
6	0.05	1.00

Solution: The random variable, X, takes on the values of x indicated in the first column, and the probability function, P(x), is defined in the second column. The third column contains the cumulative distribution, F(x). This model could be used for planning the inventory of cars. For example, if there are only four cars in stock, Stetson Motors could satisfy customers’ needs for a car 95% of the time. But if only two cars are in stock, then 35% [(1 –0.65)x100%] of the customers would not have their needs satisfied.

Derived Relationship Between Probability Function and Cumulative Probability Function

Let X be a random variable with probability function P(x) and cumulative probability function
F(). Then we can show that

where the notation implies that summation is over all possible values of x that are less than or equal to.

 Derived Properties of Cumulative Probability Functions for Discrete Random Variables
Let X be a discrete random variable with cumulative probability function F(). Then we can show that
 

Properties of Discrete Random Variables

(a) Expected Value of a Discrete Random Variable
The expected value, E(X), of a discrete random variable X is defined as

                            

where the notation indicates that summation extends over all possible values of x.

The expected value of a random variable is also called its mean and is denoted  .

Example: Suppose that the probability function for the number of errors, X, on pages from business textbooks is

                P(0) = 0.81       P(1) = 0.17         P(2) = 0.02

Find the mean number of errors per page.

Solution: We have

                
From this result it is concluded that over a large number of pages, the expectation would be to find an average of 0.21 error per page.

The concept of variance can be very useful in comparing the dispersions of probability distributions. Consider, for example, viewing as a random variable the return over a year on an investment. Two investments may have the same expected returns but will still differ in an important way if the variances of these returns are substantially different from the mean are more likely than if the variance of returns is small. In this context, then, variance of the return can be associated with the concept of the risk of an investment—the higher the variance, the greater the risk.

Taking the square root of the variance to obtain the standard deviation yields a quantity in the original units of measurement.

Example: An automobile dealer calculates the proportion of new cars sold that have been returned various numbers of times for the correction of defects during the warranty period. The results are shown in the table.

Number of returns	0	1	2	3	4
Proportion	0.28	0.36	0.23	0.09	0.04

(a)    Find the mean number of returns of an automobile for the corrections for defects during the warranty period.
(b)    Find the variance of the number of returns of an automobile for corrections for defects during the warranty period.

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