78
1 Mathematical Physics
(c) < x
2
>=
x
2 e
−m m
x
x!
=
[x(x − 1) + x]
e
−m m
x
x!
=
∞
x=0
e
−m m
x
(x − 2)!
+
∞
x=0
x e
−m m
x
x!
= e
−m
m
2
+
m
3
1!
+
m
4
2!
+ · · ·
+ m
= m
2 e
−m e
m
+ m = m
2
+ m
σ
2
=< (x − ¯
x)
2
>=< x
2
> −2 < x > ¯
x+ < ¯
x >
2
=< x
2
> −m
2
σ
2
= m or σ =
√
m
(d) P m−1 =
e
−m m
m−1
(m − 1)!
=
e
−m m
m
(m − 1)!m
=
e
−m m
m
m!
= P m
That is the probability for the occurrence of the event at x = m − 1 is
equal to that at x = m
(e) P x−1 =
e
−m m
x−1
(x − 1)!
=
e
−m m
x
x!
x
m
=
x
m
P x
P x+1 =
e
−m m
x+1
(x + 1)!
=
m e
−m m
x
x!(x + 1)
=
m
x + 1
P x
1.94 (a) (q + p)
N
= q
N
+ Nq
N −1 P +
N (N − 1)q
N −2
2!
P
2
+ · · ·
N !
x!(N − x)!
P
x q
N −x
+ · · · P
N
=
N
x=0
N !
x!(N − x)!
P
x q
N −x
= 1(∵ q + p = 1)
(b) We can use the moment generating function M x (t) about the mean μ
which is given as
M x (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 that μ n is the coefficient of
t
n
n!