1.3 Solutions 81
(b)B(x)=N!pxqN−x
x!(N−x)!=N(N−1)...(N−x−1)px(1−p)N−x
x!=N(N−1)...(N−x+1)(Np)x(1−p)N−x
Nxx!=mx
x!(
1 −
1
N
)(
1 −
2
N
)
···
(
1 −
x− 1
N)
(1−p)N−x=
mx
x!(
1 −N^1
)(
1 −^2 N
)
···
(
1 −x−N^1)
(1−p)N
(1−p)x
The poisson distribution can be deduced as a limiting case of the bino-
mial distribution, for those random processes in which the probability of
occurrence is very small,p1, while the number of trialsNbecomes
very large and the mean valuem=pnremains fixed. ThenmNand
xN, so that approximately
(1−p)N−x≈e−p(N−x)≈e−pN=e−m
ThusB(x)→P(x)=e−mmx
x!1.97 S=(g−b)±√
g
tg+
b
tbt=tb+tg=constant
tg=t−tbσ^2 =σg^2 +σb^2 =g
t−tb+
b
tb
must be minimum. Therefore,∂(σ(^2) )
∂tb =^0
g
(t−tb)^2
−
b
tb^2= 0
tb^2
(t−tb)^2=
b
g→
tb
tg=
√
b
g1.98 A=A 0 e
−λtln(
A 0
A
)
=λty=λt
S=y−λt