concerned with finding the probability that the variable assumes any value in
an interval between two specific pointsaandb. The probability that a contin-
uous variable assumes a value between two pointsaandbis the area under the
graph of the density curve betweenaandb; the vertical axis of the graph rep-
resents the densities as defined in Chapter 2. The total area under any such
curve is unity (or 100%), and Figure 3.4 shows the standard normal curve with
some important divisions. For example, about 68% of the area is contained
withinG1:
Prð 1 <z< 1 Þ¼ 0 : 6826and about 95% withinG2:
Prð 2 <z< 2 Þ¼ 0 : 9545More areas under the standard normal curve have been computed and are
available in tables, one of which is our Appendix B. The entries in the table of
Appendix B give the area under the standard normal curve between the mean
(z¼0) and a specified positive value ofz. Graphically, it is represented by the
shaded region in Figure 3.5.
Using the table of Appendix B and the symmetric property of the standard
normal curve, we show how some other areas are computed. [With access to
some computer packaged program, these can be obtained easily; see Section
3.5. However, we believe that these practices do add to the learning, even
though they may no longer be needed.]
Figure 3.4 Standard normal curve and some important divisions.124 PROBABILITY AND PROBABILITY MODELS