Page 218 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 201

of no surprise. As the number of steps increases, it is expected that position of the
particle becomes normally distributed in the limit.

7.2.2 PROBABILITY TABULATIONS

Owing to its importance, we are often called upon to evaluate probabilities
associated with a normal random variable X : N(m, 2 ), such as

" 2 #
b
Z
1 x m†
P a < X b†ˆ exp dx: 7:20†
2 † 1=2 a 2 2
However, as we commented earlier, the integral given above cannot be evaluated
by analytical means and is generally performed numerically. For convenience,
tables are provided that enable us to determine probabilities such as the one
expressed by Equation (7.20).
The tabulation of the PDF for the normal distribution with m ˆ 0 and ˆ 1
is given in Appendix A, Table A.3. A random variable with distribution N(0, 1)
is called a standardized normal random variable, and we shall denote it by U.
Table A.3 gives F U (u) for points in the right half of the distribution only (i.e.
for u 0). The corresponding values for u < 0 are obtained from the symmetry
property of the standardized normal distribution [see Figure 7.6(a)] by the
relationship

F U u†ˆ 1 F U u†: 7:21†

First, Table A.3 in conjunction with Equation (7.21) can be used to determine
<
P(a U b) for any a and b. Consider, for example, P" 1:5 < U 2:5). It is
given by

P 1:5 < U 2:5†ˆ F U 2:5† F U 1:5†:

:
The value of F U (2 5) is found from Table A.3 to be 0.9938; F U ( 1 5) is equal to

:
1 F U "1:5), with F U "1:5) ˆ 0:9332, as seen from Table A.3. Thus
P 1:5 < U 2:5†ˆ F U 2:5† ‰1 F U 1:5†Š
ˆ 0:994 1 ‡ 0:933 ˆ 0:927:

More importantly, Table A.3 and Equation (7.21) are also sufficient for
determining probabilities associated with normal random variables with arbi-
trary means and variances. To do this, let us first state Theorem 7.2.

TLFeBOOK