Variance and standard deviation of discrete random variable

While the expected value gives us an idea of the average or central value for the random variable, often we would also like to measure the dispersion or variability of the possible values of the random variable. The variance of discrete random variable X, denoted by , measures the spread of its probability distribution. In defining the variance of a random variable, a weighted average of the squares of its possible discrepancies about the means is formed; the weight associated with is the probability that the random variable takes the value x. The variance can be viewed as the average value that will be taken by the function over a very large number of repeated trials.

The mathematical expression for the variance of a discrete random variable is

.

The standard deviation, , is the positive square root of the variance.

In some particular cases, an alternative but equivalent (sometimes called shortcut formula) formula for the variance can be used:

Example:

Find the variance for the example in previous topic, for the number of cars sold per day at a car selling company.

Solution:

Let us apply .

The calculations are shown in the table 3.5:

Table3.5

x			P (x)
	0-1.50=-1.50 1-1.50=-0.50 2-1.50=0.50 3-1.50=1.50 4-1.50=2.50 5-1.50=3.50	2.25 0.25 0.25 2.25 6.25 12.25	0.18 0.39 0.24 0.14 0.04 0.01	2.25·0.18=0.4050 0.25·0.39=0.0975 0.25·0.24=0.0600 2.25·0.14=0.3150 6.25· 0.04=0.2500 12.25· 0.01=0.1225

We see that the variance for the number of cars sold per day is 1.25.

The standard deviation of the number of cars sold per day is .

Remark:

For the purpose of easier managerial interpretation the standard deviation may be preferred over the variance because it is measured in the same units as the random variable.

Example:

The following table gives the probability distribution of X.

x	0 1 2 3 4 5
P (x)	0.02 0.20 0.30 0.30 0.10 0.08

Compute the standard deviation of x.

Solution:

Let us apply equivalent (shortcut formula) formula for the variance

The following table shows all the calculations required for the computation of the standard deviation of x.

x	P (x)	xP (x)
	0.02 0.20 0.30 0.30 0.10 0.08	0.00 0.20 0.60 0.90 0.40 0.40		0.00 0.20 1.20 2.70 1.60 2.00

We perform the following steps to compute the standard deviation by shortcut formula:

Step1: Compute the mean of discrete random variable:

Step2: Compute the value of .

Step3: Substitute the values of and in the shortcut formula for the variance

=

Step4: Take positive square root of variance.

.

Example:

A farmer will earn a profit of $30 thousand in case of heavy rain next year, $60 thousand in case of a moderate rain, and $15 thousand in case of little rain. A meteorologist forecasts that the probability is 0.35 for heavy rain, 0.40 for moderate rain, and 0.25 for little rain next year. Let X be the random variable that represents next year’s profit in thousands of dollars for this farmer. Write the probability distribution of x. Find the mean and standard deviation of x. Give a brief interpretation of the values of the mean and standard deviation.

Solution: Table 3.6

x	P (x)
	0.35 0.40 0.25

The table3.6 lists the probability distribution of x

The table 3.7 shows all calculations needed for the computation of the mean and standard deviation.

Table 3.7

x	P (x)	xP (x)
	0.35 0.40 0.25	10.5 3.75		56.25

The mean of x is $38.25 thousand. The standard deviation is $18.660.

Thus, it is expected that a farmer will earn an average of $38.25 thousand profits in next year with a standard deviation of $18.660 thousand.

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