Basic Probability:
A ( B C ) ( A B) ( A C )
A ( B C ) ( A B) ( A C )
M X (0) 1
( A B ) ' A ' B '
( A B ) ' A ' B '
M aX b (t ) etb M X (at )
P ( X 1 x1 , X 2 x2 ,..., X k xk )
M X (t ) E (etX ) etX p ( x )
MX
(n)
(0) x n p ( x) E ( X n )
n!
x
x
x
p1 1 p2 2 ... pk k
x1 !x2 !
... xk !
P ( A B ) P ( A) P ( B ) P ( A B )
P ( A B C ) P ( A) P ( B ) P (C )
P( A B) P ( A C ) P ( B C ) P ( A B C )
Continuous Random Variables
n
n!
r !(n r )!
r
b
P (a X b) f ( x)dx
a
P( A B)
P( B)
P ( A B ) P( A / B ) P ( B )
P ( A '/ B) 1 P ( A / B)
P ( A B / C ) P ( A / C ) P ( B / C ) P( A B / C )
P ( A) P ( A / B ) P ( B ) P( A / B ') P ( B ')
P ( E ) P ( A1 E ) P ( A2 E ) ... P ( An E )
P( A / B)
f ( x)dx 1
F ( x) P ( X x)
x
F ( x)
f (u )du
F '( x) f ( x )
P (a X b) F (b) F (a )
lim F ( x ) 1
x
Independence:
P ( A / B ) P ( A)
P ( A B ) P( A) P ( B )
Discrete Random Variables
a (1 r n 1 )
2
n
a ar ar ... ar
1 r
a
a ar ar 2 ... ar n ...
1 r
E ( X ) x p ( x) Mean
E ( aX ) aE ( X )
E ( aX b) aE ( X ) b
V ( X ) ( x ) 2 p ( x) E ( x ) 2
V (X ) 2
V (aX ) a 2V ( X )
V (aX b) a 2V ( X )
V ( X ) E ( X 2 ) E ( X )2
lim F ( x) 0
x
d
d
P (T t ) P(T t ) or F '( x) S '( x)
dt
dt
E( X )
x f ( x)dx
E g ( X )
g ( x) f ( x)dx
M X (t ) E (e )
tX
e
x e
n ax
dx
0
cx
xe dx
xecx ecx
2
c
c
xn
0
x e ax
lim
n!
a n 1
tx
f ( x)dx
E E ( X / Y ) E ( X )
Multivariate Distributions
E E (Y / X ) E (Y )
PX ( x ) p ( x, y )
V ( X / Y ) E ( X 2 / Y y) E ( X / Y y)
y
PY ( y ) p( x, y )
V ( X ) E V ( X / Y ) V E ( X / Y )
x
b d
V (Y ) E V (Y / X ) V E (Y / X )
a c
M X ,Y (t1 , t2 ) E (et1 X t2Y )
P (a X b, c Y d ) f ( x, y )dydx
f X ( x)
M X ,Y (t1 , 0) E (et1 X ) M X (t1 )
f ( x, y )dy
fY ( y )
M X ,Y (t1 , t2 )t1 t2 0
t1
E (Y )
M X ,Y (t1 , t2 )t1 t2 0
t2
f ( x, y )dx
P( X x / Y y ) p( x / y )
f ( x / Y y) f ( x / y)
p ( x, y )
pY ( y )
f ( x, y )
fY ( y )
E ( X / Y y ) x p ( x / y )
E ( X / Y y)
M X ,Y (t1 , t2 )t1 t2 0
t1t2
p ( x / y ) p X ( x)
x f ( x / y )dx
p ( y / x) pY ( y )
f ( x / y ) f X ( x)
f ( x)
P ( A)
f ( y / x ) fY ( y )
f ( x , y ) f X ( x ) f Y ( y )
E ( X Y ) E ( X ) E (Y )
Cov( X , Y ) E ( XY ) E ( X ) E (Y )
Cov( X , X ) V ( X )
Cov(Y , Y ) V (Y )
Cov( X , k ) 0
Cov(aX , bY ) abCov( X , Y )
Cov(W X , Y Z ) Cov(W , Y ) Cov(W , Z )
Cov( X , Y ) Cov( X , Z )
V ( X Y ) V ( X ) V (Y ) 2Cov( X , Y )
V (aX bY c ) a 2V ( X ) b 2V (Y ) 2abCov( X , Y )
V (aX bY cZ ) a 2V ( X ) b 2V (Y ) c 2V ( Z )
2 abCov ( X , Y ) acCov ( X , Z ) bcCov(Y , Z )
Correlation Coefficient: XY
E ( XY )
Independence:
x
f ( x / A)
E( X )
Cov( X , Y )
V ( X ) V (Y )
E ( XY ) E ( X ) E (Y )
Cov( X , Y ) 0
V ( X Y ) V ( X ) V (Y )
M X Y (t ) M X (t ) M Y (t )
Sums of Distributions
Single
Bernoulli
Binomial
Poisson
Geometric
Negative Binomial
Normal
Exponential
Gamma
Chi-Square