Page 346 - Numerical Methods for Chemical Engineering

Important probability distributions 335

Next, we take the natural logarithm and apply Stirling’s approximation,

n!
ln = ln(n!) − ln[(n − ζ)!] ≈ n ln n − n − (n − ζ) ln(n − ζ) + n − ζ
(n − ζ)!
≈ n ln n − n − (n − ζ) ln(n − ζ) + n − ζ ≈ ζ ln n (7.96)
Hence,

n! ζ
≈ n (7.97)
(n − ζ)!
and the binomial distribution reduces to the Poisson distribution
(pn) ζ −pn
P(ζ; n, p) = e (7.98)
ζ!
with the normalization
n n ζ
−pn (pn) −pn +pn
P(ζ; n, p) = e = e e = 1 (7.99)
ζ!
ζ=0 ζ=0
and the same expectation and variance as the binomial distribution,

E(ζ) = pn var(ζ) = np(1 − p) (7.100)
The statistics toolkit offers several functions for the Poisson distribution, whose value is
returned by poisspdf. The cumulative distribution and its inverse are returned by poisscdf
and poissinv. To fit a Poisson distribution to a data set use poissfit; the mean and standard
deviation are returned by poissstat. Random numbers are returned by poissrnd.
A classic application of the Poisson distribution is the question

If we buy a very large number n of lottery tickets, each with a very small probability p of winning,
what is the probability that we will have bought at least one winner?
Applying the Poisson distribution, this probability is
n
−pn
P(ζ ≥ 1; n, p) = P(ζ; n, p) = 1 − P(0; n, p) = 1 − e (7.101)
ζ=1
The Poisson distribution finds common use in the study of many physical phenomena. For
example, consider the case of anionic living polymerization. Using an initiator such as
n-butyl lithium that forms a carbanion, we can polymerize vinyl monomers (Figure 7.10).
Initiation is rapid, so that we start growing each chain at the same time. In a very small
time interval t, the probability that we add a monomer unit to a specific chain during
this interval is k p [M]( t), where k p is a propagation rate constant and [M] is the monomer
concentration. To account for the changing monomer concentration, we define a scaled
time
t
'
[M](t )dt (7.102)

τ = k p
0
and take n steps forward in τ, each of duration τ. The probability of adding a monomer