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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 ... pk k x1 !x2 !  ... xk !

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 t1t2

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

Multiple Binomial Binomial Poisson Negative Binomial Negative Binomial Normal Gamma Gammas Chi-Square

2

If X,Y have joint distribution which is uniform, 1 then pdf is Area of R



E ( X )   ( x  d ) f x ( x) dx  (u  d )  1  Fx (u )  u

Area of A then P(A) = Area of R

u

   1  Fx ( x)  dx d

Bivariate Normal Cov( X , Y ) ( X E ( X )) V (X )

E (Y / X )  E (Y ) 

V (Y / X )  V (Y ) (1  2 XY ) Bernoulli = Binomial with only 1 trial Coefficient of Variation =

 

Memoryless Exponential, Uniform, Geometric Combine Normals Add/Subtract Means Add variances Deductibles and Policy Limits  0  xd

Deductible: Y  

xd xd





d

d

E ( X )   ( x  d ) f x ( x )dx    1  F ( x) dx  x  u

Policy Limit: Y  

xu xu

u



u

0

u

0

E ( X )   xf x ( x)dx   u f x ( x)dx   xf x ( x)dx  u  1  Fx (u )  dx u

   1  Fx ( x ) dx 0

Deductibles and Policy Limits