p1 soa exam study manual

A&J Flashcards for Exam P/1 1st Edition

Alvin Soh

Outline A) Combinatorial Analysis Analysis B) Basic Probability and Set Set Theory C) Conditional Probability and Bayes Formula D) Discrete Random Variables E) Discrete Probability Distribution F) Continuous Random Variables G) Continuous Probability Distribution H) Jointly Distributed Random Random Variables Variables with its its Applications Applications I)

Jensen Inequality and Chebyshev’s Inequality

J)

Application of Probability in Risk Management

A&J © 2010. These flashcards were purchased by [Candidate P].

Basic Probabilities and Set Theory

Distributive Law



 A  B  C  AC  BC  AB  C   AC    BC 



De Morgan’s Law



C

n   C E E   i   i i 1  i 1  n

C

n   C E E   i   i i 1  i 1  n



Law of Total Probability



n  A  n  AB   n  AC   n  AD  

Pr  A  Pr  AB   Pr  AC   Pr  AD  


Conditional Probability and Bayes Formula

Bayes Formula


Conditional Probability and Bayes Formula

Pr  A | B  

Pr  AB  Pr  B 



Pr  B | A Pr  A  Pr  B | A Pr  A  Pr  B | AC  Pr  A C 


Discrete Random Variables

Characteristics of Probability Mass Function



f ( x)  Pr( X  x) 2. f ( x)  0, x   1.

3.

 f ( x)  1 x



th

k

moment



 

E X k 

 x: f ( x ) 0

xk f ( x) 



xk Pr  X  x

x: f ( x ) 0


Discrete Probability Distribution

If X follows binomial distribution, find: 1. Probability Mass Function 2. Expected Value 3. Variance


Discrete Probability Distribution

The distribution is commonly represented by

 n  x n x X ~ Bin  n, p  , with Pr  X  x     p 1  p  , where  x  x  0,1,..., n , n and p are parameters for this distribution. E  X   np

Var  X   np 1  p 


Continuous Random Variable

Transformation of Random Variables using “CDF” Method



Pr Y  g  X   Pr  X  g 1 Y  or Pr  X  g 1 Y   depending on the function, g



Transformation of Random Variables using “Inverse Transformation” Method



If Y

 g  X  , g '  X   0 ,  x   then the p.d.f of Y is

fY  y   f X  h  y   h  y  , where h  y   g 1  y  '


Continuous Probability Distribution

If X follows Exponential distribution, find: 1. 2. 3. 4.

Probability Density Function Expected Value Variance Moment-Generating Function


Continuous Probability Distribution

 1  x x0  e , f  x     0 , elsewhere  x

F  x  

1

 



y

e  dy  1  e



x 

0

E  X   

Var  X    2

M X  t  

1 1   t



Relationship between the expected payment and the expected payment with deductible



E  X   E  X ^ d   E  X  d   whereby,



d

d







0

d

0

E  X ^ d   xf ( x)dx  d f ( x) dx  S  x  dx 

E  X  d   



  x  d  f ( x)dx   S  x dx d

d



Coinsurance



E  Payment   E  1    X   1    E  X 




p1 soa exam study manual

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