Statistical Physics & Thermodynamics
Prof. Dr. Haye Hinrichsen, Stefan Beyl, Laura Lauerbach, Lisa Morgan, Manuel Schrauth WS 14/15
SAMPLE SOLUTION
Exercise 1 Binomial distribution (4P)
The binomial distribution reads
N k
PN,p (k) =
p (1 − p)N −k .
k
−1
(a) Use the relation Nk = Nk Nk−1
to derive a recursion relation for the non-centralized
moments mn (N, p) = hk n iN,p of the binomial distribution. This recursion relation
should have the following form:
mn (N, p) = some function of m0 (N − 1, p), m1 (N − 1, p), . . . , mn−1 (N − 1, p)
(b) Apply this recursion relation to compute the first four non-centralized moments
m0 , . . . , m3 explicitly.
(c) Compute the moment-generating function Mn,p (t) = hekt iN,p of the binomial distribution analytically (please provide a complete proof).
(d) Compute the first three cumulants κ0 , . . . , κ3 explicitly from the cumulant-generating
R
function KN,p (t) = ln MN,p (t). You are encouraged to use Mathematica
or similar
computer algebra software.
Good luck!
Sample solution for exercise 1:
(a)
n
mn (N, p) = hk iN,p
N
X
N
X
N k
N −k
nN N − 1
=
k
p (1 − p)
=
k
pk (1 − p)N −k
k k−1
k
k=0
k=1
N
−1
X
N − 1 k+1
= N
(k + 1)n−1
p (1 − p)N −k−1
k
n
k=0
Now we expand (k + 1)n−1 =
mn (N, p) = N p
Pn−1
n−1
X
q=0
= Np
n−1
X
q=0
q=0
n−1
q
n−1
q
k q to obtain
NX
−1
k=0
kq
N −1 k
p (1 − p)N −1−k
k
n−1
mq (N − 1, p)
q
(b) The zeroth moment is the norm which anchors the recursion:
N X
N k
m0 (N, p) =
p (1 − p)N −k = (p + 1 − p)N = 1
k
k=0
The first moment is the mean:
m1 (N, p) = N pm0 (N, p) = N p
The second and the third moment can be computed similarly, giving:
m2 (N, p) = N p (N p − p + 1)
m3 (N, p) = N p N 2 p2 − 3N p2 + 3N p + 2p2 − 3p + 1 .
(c)
kt
M (t; N, p) = he iN,p
N
X
N k
=
p (1 − p)N −k
e
k
k=0
N X
N
=
(p et )k (1 − p)N −k = (pet + (1 − p))N .
k
kt
k=0
(d) The Mathematica command
Series[Log[M[t]], {t, 0, 3}]
gives the output:
1
1
N pt + t2 N p − N p2 + t3 2N p3 − 3N p2 + N p + O t4
2
6
From this we can read off the cumulants
κ0 = 0,
κ1 = N p,
κ2 = N p(1 − p),
κ3 = N p(1 − p)(1 − 2p)
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