Biomedical Engineering Reference
In-Depth Information
Fig. 11.3
Comparison of binomial (histogram) and normal
(solid line) distributions, having the same means and standard
deviations. The ordinate in each panel gives the probability
P
n
for the former and the density
f
(
x
) [Eq. (11.37)] for the latter, the
abscissa giving
n
or
x
. (Courtesy James S. Bogard.)
standard normal distribution, having zero mean and unit standard deviation, is
obtained by making the substitution
x
-
µ
z
=
.
(11.39)
σ
Equation (11.38) then becomes (
d
x
=
σ
d
z
)
z
2
1
√
2
e
-
z
2
/2
d
z
.
P
(
z
1
≤
z
≤
z
2
) =
(11.40)
π
z
1
Table 11.1 lists values of the integral,
z
0
1
√
2
e
-
z
2
/2
d
z
,
(11.41)
P
(
z
≤
z
0
)
=
π
-
∞
giving the probability that the normal random variable
z
has a value less than or
equal to
z
0
. This probability is illustrated by the shaded area under the standard
normal curve, as indicated at the top of Table 11.1. The following example illus-
trates the use of the table.
Example
Repeated counts are made in 1-min intervals with a long-lived radioactive source.
The observed mean value of the number of counts is 813, with a standard devia-
