78 1 Mathematical Physics
(c)<x^2 >=
∑
x^2
e−mmx
x!
=
∑
[x(x−1)+x]
e−mmx
x!
=
∑∞
x= 0
e−mmx
(x−2)!
+
∑∞
x= 0 xe
−mm
x
x!
=e−m
(
m^2 +
m^3
1!
+
m^4
2!
+···
)
+m
=m^2 e−mem+m=m^2 +m
σ^2 =<(x−x ̄)^2 >=<x^2 >− 2 <x>x ̄+< ̄x>^2 =<x^2 >−m^2
σ^2 =morσ=
√
m
(d)Pm− 1 =
e−mmm−^1
(m−1)!
=
e−mmm
(m−1)!m
=
e−mmm
m!
=Pm
That is the probability for the occurrence of the event atx=m−1is
equal to that atx=m
(e)Px− 1 =
e−mmx−^1
(x−1)!
=
e−mmx
x!
x
m
=
x
m
Px
Px+ 1 =
e−mmx+^1
(x+1)!
=
me−mmx
x!(x+1)
=
m
x+ 1
Px
1.94 (a)(q+p)N=qN+NqN−^1 P+
N(N−1)qN−^2
2!
P^2
+···
N!
x!(N−x)!
PxqN−x+···PN
=
∑N
x= 0
N!
x!(N−x)!
PxqN−x=1(∵q+p=1)
(b) We can use the moment generating functionMx(t) about the meanμ
which is given as
Mx(t)=Ee(x−μ)t
=E
[
1 +(x−μ)t+(x−μ)^2
t^2
2!
+···
]
= 1 + 0 +μ 2
t^2
2!
+μ 3
t^3
3!
+···
So thatμnis the coefficient oft
n
n!