q

X

xt =

(Uk1 cos(2 !kt) + Uk2 sin(2 !kt))

k=1

for k = 1; : : : ; q, are independent zero-mean rv's with variances 2

Uk1; Uk2, q

2 cos(2 ! h)

k

P t

k

k=1

k

(h) =

k=1

q

k

(0) = E x2

=

2

Spectral

representation of a periodic process

P

(h) = 2 cos(2 !0h)

2

2

=

e 2 i!0h

+

e2 i!0h

2

2

Spectral distribution function

8 2

!0< !0

F (!) =

=2 !

>

0

! <

!

2

<

!

!0

>

F (1) = F ( 1=2) = 0

:

F (1) = F (1=2) = (0)

Spectral density

1

1

1

X

(h)e 2 i!h

!

f(!) =

2

2

h=1

1

1=2

2 i!h

f(!) d! h = 0; 1; : : :

Needs Ph=1 j (h)j < 1 =)

(h) = R 1=2 e

f

x

(!) = 2

j (e 2 i!)j2

w j (e 2 i!)j2

p

q

kzk

where (z) = 1 Pk=1

kzk and (z) = 1 + Pk=1

n

Xi

d(!j) = n 1=2

xte 2 i!jt

=1

Fourier/Fundamental frequencies

!j

= j=n

Inverse DFT

n 1

Xj

xt = n 1=2

d(!j)e2 i!jt

=0

Periodogram

I(j=n) = jd(j=n)j2

Scaled Periodogram

P (j=n) =

4

I(j=n)

n

2

n

2

2

n

2

X

Xt

+ n

=

n t=1 xt cos(2 tj=n!

=1 xt sin(2 tj=n!

( + 1) = ( )

> 1

(n) = (n 1)!

n 2 N

p

(1=2) =

Ordinary: B(x; y) = B(y; x) =

Z0

1

tx 1(1 t)y 1 dt = (x + y)

(x) (y)

Incomplete: B(x; a; b) = Z0x ta 1(1 t)b 1 dt

Regularized incomplete:

a+b 1

B(x; a; b)

a;b

I (a; b) =

=2N

Xj

(a + b 1)!

xj(1

x)a+b 1 j

x

B(a; b)

j!(a + b

1

j)!

26

Stirling numbers, 2nd kind

(0

k

k

k

1

0

else

n = k n

1 + n 1

1 k n

n = 1 n = 0

Partitions

n

Pn+k;k =

Xi

k > n : Pn;k = 0

n 1 : Pn;0 = 0; P0;0 = 1

Pn;i

=1

Balls and Urns

f : B ! U

D = distinguishable, :D = indistinguishable.

jBj = n, jUj = m

f arbitrary

f injective

f surjective

f bijective

(0

else

m

(0

else

B : D; U : D

mn

mn

m

n

m! n

n! m = n

B : :D; U : D

n

n

m

1

(0

else

1

m = n

m + n 1

m

n 1

:

m

k

(0

else

m

(0

else

B : D; U : D

k=1

1 m n

1

m = n

X

B : D; U : D

m

P

1 m n

P

1

m = n

X

:

:

n;k

(0

else

n;m

(0

else

k=1