Q1
2R 2p
(n 1) n # (n)
(n 0) 3
`
0
x n1 # ex dx (n)
v 22pq
(q p) 3
`
0
exp(wz) # Q(12pz,12qz )dz
u 2u2 v2 2u2 v2 (u 2u2 v2 2p)
DEFINITE INTEGRALS 519
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APPENDIX D
PROBABILITY FUNCTIONS
In this section, we list some of the well-known random-variable functions along with their mean
value and variance.
Poisson
Binomial
Uniform
Gaussian
s2x
s2 x m
(xm)2
2s
2 pX (x)
1
s22p
e
s2x
(b a)2
12
x
(a b)
2
pX (x) c 1
b a
a x b
0 otherwise
s2x
na (1 a) x na
(x 0, 1, c, n) PX (x) an
xb # ax # (1 a)nx
s2x
K x K
PX (x) eK Kx
x!
521
Rayleigh
Exponential
s2x
1
K2 x
1
K
(x 0) pX (x) K # eKx
s2x
a2
p
2 bs2 x sAp
2
(x 0)
x2
2s
2 pX (x)
x
s2
# e
522 APPENDIX D
APPENDIX E
SERIES AND SUMMATIONS
aK
k1
k3
K2(K 1)2
4
aK
k1
k2
K(K 1)(2K 1)
6
aK
k1
k
K(K 1)
2
Io(x) a`
k0
x2k
22k (k!)2
Jo(x) a`
k0
(1)k x2k
22k (k!)2
e a`
k0
1
k!
sin (px)
px 1
1
3!
(px)2
1
5!
(px)4 c
ln (1 x) x
1
2
x2
1
3
x3 c
ax 1 x ln (a)
1
2
[x ln (a)]2 c
ex 1 x
1
2!
x2
1
3!
x3 c
(x 1) 1 a`
k1
n(n 1)(n 2)c(n k 1)
k!
xk
(1 x)n 1 nx
n(n 1)
2!
x2 c
tan (x) x
1
3
x3
2
15
x5
17
315
x7 c
cos (x) 1
1
2!
x2
1
4!
x4
1
6!
x6 c
sin (x) x
1
3!
x3
1
5!
x5
1
7!
x7 c
523
Cauchy-Schwarz Inequality
Minkowski??™s Inequality
Holder??™s Inequality
with a1
P
1
Q 1b aN
k1
akbk aaN
k1
aPk
b1/P
# aaN
k1
bQk
b1>Q
aaN
k1
(ak bk)Pb1/P
aaN
k1
aPk
b1/P
aaN
k1
bPk
b1/P
aaN
k1
akbkb2
aaN
k1
a2k
b # aaN
k1
b2k
b
a`
k1
axk1
1
1 x
aN
k1
axk1
a(1 xN)
1 x
aK
k0
xk
xK1 1
x 1
524 APPENDIX E
APPENDIX F
LINEAR ALGEBRA
Let A be an n n matrix, then the following properties can be mentioned for determinants, where
det .
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