Introduction to Probability and Statistics for Engineers and Scientists

5.5Normal Random Variables 171

−x 0 x

P {Z < −x} PP {Z > x}

FIGURE 5.8 Standard normal probabilities.

It remains for us to compute (x). This has been accomplished by an approximation
and the results are presented in Table A1 of the Appendix, which tabulates (x) (to a 4-digit
level of accuracy) for a wide range of nonnegative values ofx. In addition, Program 5.5a of
the text disk can be used to obtain (x).
While Table A1 tabulates (x) only for nonnegative values ofx, we can also obtain
(−x) from the table by making use of the symmetry (about 0) of the standard normal
probability density function. That is, forx>0, ifZrepresents a standard normal random
variable, then (see Figure 5.8)

(−x)=P{Z<−x}

=P{Z>x} by symmetry

= 1 − (x)

Thus, for instance,

P{Z<− 1 }= (−1)= 1 − (1)= 1 −.8413=.1587

EXAMPLE 5.5a IfXis a normal random variable with mean μ = 3 and variance
σ^2 =16, find

(a)P{X< 11 };

(b)P{X>− 1 };

(c) P{ 2 <X< 7 }.

SOLUTION

(a) P{X< 11 }=P

{

X− 3

4

<

11 − 3

4

}

= (2)

=.9772

(b) P{X>− 1 }=P

{

X− 3

4

>

− 1 − 3

4

}

=P{Z>− 1 }