EXAMPLE 4.35 Find the Mean and the Variance

CASE 4.2 In Case 4.2 (pages 210–211), we saw that the distribution of the daily demand of transfusion blood bags is

Bags used 0 1 2 3 4 5 6
Probability 0.202 0.159 0.201 0.125 0.088 0.087 0.056
Bags used 7 8 9 10 11 12
Probability 0.025 0.022 0.018 0.008 0.006 0.003

We can find the mean and variance of by arranging the calculation in the form of a table. Both and are sums of columns in this table.

0 0.202 0.00
1 0.159 0.159
2 0.201 0.402
3 0.125 0.375
4 0.088 0.352
5 0.087 0.435
6 0.056 0.336
7 0.025 0.175
8 0.022 0.176
9 0.018 0.162
10 0.008 0.080
11 0.006 0.066
12 0.003 0.036

230

We see that . The standard deviation of is . The standard deviation is a measure of the variability of the daily demand of blood bags. As in the case of distributions for data, the connection of standard deviation to probability is easiest to understand for Normal distributions (for example, 68–95–99.7 rule). For general distributions, we are content to understand that the standard deviation provides us with a basic measure of variability.