Biomedical Engineering Reference
In-Depth Information
Example
Repeat the last example by using Poisson statistics to approximate the binomial dis-
tribution.
Solution
(a) As before, the mean disintegration rate is the given activity, which we write as
µ
=
37 s
-1
.
(b) The standard deviation [Eq. (11.34)] is
σ
=
√
µ
=
√
37
6.08 s
-1
,
=
(11.35)
as compared with 6.09 s
-1
found before [Eq. (11.28)].
(c) The probability of exactly 40 disintegrations occurring in a given second is, by
Eq. (11.33),
37
40
e
-37
40!
P
40
=
=
0.0559,
(11.36)
in close agreement with the value 0.0561 found before. (Lack of exact agreement
to several significant figures between the results found here with the binomial and
Poisson distributions can be attributed to round-off.)
Like the binomial distribution, the distribution (11.33) can be derived in its own
right for a Poisson process.
3)
The conditions required are the following:
1. The number of successes in any one time interval is
independent of the number in any other disjoint time interval.
(The Poisson process has no memory.)
2. The probability that a single success occurs in a very short time
interval is proportional to the length of the interval.
3. The probability that more than one success will occur in a very
short time interval is negligible.
The Poisson process can describe such diverse phenomena as the number of traffic
accidents that occur during August in a certain county, the number of eggs laid
daily by a brood of hens, and the number of cosmic rays registered hourly in a
counter. The events occur at random, but at an expected average rate. Generally,
the Poisson distribution describes the number of successes for any random process
whose probability is small (
p
1
) and constant.
Figure 11.1 shows a comparison of the binomial and Poisson distributions. In all
panels, the mean,
µ
= 10
, of both distributions is kept fixed; the probability of suc-
cess
p
and sample size
N
are varied between panels. Since the mean is the same,
the Poisson distribution is the same throughout the figure. Both distributions are
asymmetric, favoring values of
n
≤
µ
. As pointed out after Eq. (E.19), although the
binomial probability
P
n
=
0
when
n
>
N
, the Poisson
P
n
are never exactly zero. (In
the upper left-hand panel of Fig. 11.1 when
n
>15
, for example, it can be seen
3 However, we shall not carry out the derivation
of Eq. (11.33) from the postulates.
