Biomedical Engineering Reference
In-Depth Information
11.4
The Binomial Distribution
We summarize the results that describe the Bernoulli process of radioactive decay
and generalize Eq. (11.11) for any initial number
N
of identical radioactive atoms.
The probability that exactly
n
will disintegrate in time
t
is
N
n
p
n
q
N
-
n
.
P
n
=
(11.13)
Here
p
and
q
are defined by Eqs. (11.1) and (11.2). Since the
P
n
are just the terms in
the binomial expansion and since
p
+
q
=
1
, the probability distribution represented
by Eq. (11.13) is normalized; that is,
N
(
p
+
q
)
N
P
n
=
=
1.
(11.14)
n
=
0
The function defined by Eq. (11.13) with
p
+
q
= 1
is called the binomial distribu-
tion and applies to any Bernoulli process. Besides radioactive decay, other familiar
examples of binomial distributions include the number of times “heads” occurs
when a coin is tossed
N
times and the frequency with which exactly
n
sixes occur
when five dice are rolled. The binomial distribution finds widespread industrial
applications in product sampling and quality control.
The expected, or mean, number of disintegrations in time
t
is given by the aver-
age value
µ
of the binomial distribution (11.13):
n
N
n
p
n
q
N
-
n
.
N
N
µ
≡
nP
n
=
(11.15)
n
=
0
n
=
0
This sum is evaluated in Appendix E. The result, given by Eq. (E.4), is
µ
=
Np
.
(11.16)
Thus, the mean is just the product of the total number of trials and the probability
of the success of a single trial.
Repeated observations of many sets of
N
identical atoms for time
t
is expected to
give the binomial probability distribution
P
n
for the number of disintegrations
n
.
The scatter, or spread, of the distribution of
n
is characterized quantitatively by
its variance
σ
2
or standard deviation
σ
, defined as the positive square root of the
variance. The variance is defined as the expected value of the squared deviation
from the mean of all values of
n
:
N
2
)
2
P
n
.
σ
≡
(
n
-
µ
(11.17)
n
=
0
As shown in Appendix E, [Eq. (E.14)], the standard deviation is given by
σ
=
Npq
.
(11.18)
