A&J PFlashcards JosephShackelford

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A&J Flashcards for

Exam P/11st Edition

Alvin Soh, ASA, CERA



Outline

A&J © 2012. These flashcards were purchased by Joseph Shackelford.

A) Combinatorial Analysis

B) Basic Probability and Set TheoryC) Conditional Probability and Bayes Formula

D) Discrete Random Variables

E) Discrete Probability Distribution

F) Continuous Random Variables

G) Continuous Probability Distribution

H) Jensen Inequality and Chebyshev’s Inequality

I) Application of Probability in Risk Management



Combinatorial Analysis


Factorials





! ( 1)( 2)...(2)(1) , where is an integern n n n n





If M represents the number of arrangements for n objects, where n 1 objects alike, n 2 objects alike,…, n r objects alike, M =





1 2

!

! ! !r

n

n n n





Multinomial Coefficient





The number of division of objects into distinctive groups of size

is given by:

n r

1 2, , r n n n

1 2 1 2

!

, , , ! ! !r r

n n

n n n n n n



Basic Probabilities and Set Theory


Commutative Law





A B B A

AB BA





Associative Law





A B C A B C

AB C A BC





Distributive Law





A B C AC BC

AB C AC BC





De Morgan’s Law





1 1

C n n

C

i i

i i

E E

1 1

C n n

C

i i

i i

E E





Law of Total Probability





Pr Pr Pr Pr A AB AC AD

n A n AB n AC n AD





Relationship between union of two sets:

n A B





n A B n A n B n AB

Pr Pr Pr Pr A B A B AB





n A B C





n A B C

n A n B n C n AB n AC n BC n ABC

Pr

Pr Pr Pr Pr Pr Pr Pr

A B C

A B C AB AC BC ABC





Definition of Probability





# of outcomes in the event

Probability that event A happens = PrTotal # of possible event

A





If event A and B are mutually exclusive , then

Pr AB Pr A B







If event A and B are independent of each other, then

Pr AB





Pr Pr Pr AB A B



Conditional Probability and Bayes Formula


Pr | A B





PrPr |

Pr

n AB AB A B

n A A

(This can be interpreted as the probability of event A happens giventhat event B has happened)





Bayes Formula





Pr Pr | PrPr |

Pr Pr | Pr Pr | PrC C

AB B A A A B

B B A A B A A



Discrete Random Variables


Characteristics of Probability Mass Function





1. ( ) Pr( ) f x X x

2. ( ) 0, f x x

3. ( ) 1 x

f x





E X

d bl





: ( ) 0 : ( ) 0

( ) Pr x f x x f x

E X xf x x X x

OR

0 0

( ) 1 ( ) , x n

E X S x F x x

Di R d V i bl





th

k moment

Di t R d V i bl





: ( ) 0 : ( ) 0

( ) Prk k k

x f x x f x

E X x f x x X x






E aX b






E aX b E aX b

for any constants a and b as well as random variable X .






E g X






: ( ) 0

( ) ( ) x f x

E g X g x f x






Moment-Generating Functions






tx tX

t

M t E e e f x






Relationship between thk moment and moment-generating

function





0

k k

X k t

d M t E X

dt





Variance, Var X





2

Var X E X E X

OR

22

Var X E X E X





Var aX b





2Var aX b a Var X





Standard Deviation, X





X

Var X

Discrete Probability Distribution




If X follows uniform distribution, find:

1. Probability Mass Function

2. Expected Value3. Variance





The distribution is commonly represented by

~ U , X a b , with 1

Pr , , , X x a b a bb a

, a and b

are the parameters for this distribution.

2

a b E X

2

12

b aVar X





If X follows binomial distribution, find:

1. Probability Mass Function2. Expected Value3. Variance






~ Bin , X n p , with Pr 1n x x

n X x p p

x

, where

0,1,..., x n , n and p are parameters for this distribution.

E X np 1Var X np p





If X follows Poisson distribution, find:

1. Probability Mass Function2. Expected Value3. Variance






~ X Poi , with Pr!

xe

X x x

,

where is the parameter for this distribution.

E X Var X





If X follows Geometric distribution, find:








~ X Geo p , with 1

Pr 1x

X p p

,

where p is the parameter for this distribution.

1

E X p

2

1Var X

p





If X follows Negative Binomial distribution, find:








~ , X NB r p , with 1

Pr 11

x r n

X x p pr

,

where r and p are the parameters for this distribution.

1r p

E X p

2

1r pVar X

p





If X follows Hypergeometric distribution, find:








~ , , X HyperGeo N n r , with Pr

N r r

n x x X x

N

n

,

where N , n , and r are the parameters for this distribution.

r

E X n N

1

1

r r N nVar X n

N N N

Continuous Random Variable




Characteristics of Continuous Random Variables





1. Pr 0 X x f x

2. Pr

b

a

a X b f x dx

3. 1 f x dx





Expected Value and Variance





E X xf x dx

OR 0

E X S x dx

2 2

E X x f x dx

22Var X E X E X





Moment-Generating Function of a Continuous Random Variable





tX tX M t E e e f x dx





Transformation of Random Variables

using

“CDF” Method





1 1Pr Pr PrY g X X g Y or X g Y

depending on the function, g





Transformation of Random Variables

using

“Inverse Transformation” Method





If Y g X , '0g X , x then the p.d.f of Y is

'

Y X f y f h y h y , where 1

h y g y





Order Statistics



Continuous Probability Distribution




If X follows Uniform distribution, find:

1. Probability Density Function



1

, a x bf




,

0 , elsewhere

f x b a

0 ,

,

0 ,

x a

x aF x a x b

b a

x b

2

a b E X 2

12

b aVar X





If X follows Beta distribution, find:







11 1 , 0 1

0 , elsewhere

x x x f x

E X

2

1Var X





If X follows Exponential distribution, find:


2. Expected Value3. Variance4. Moment-Generating Function





1, 0

0 , elsewhere

x

e x f x

0

11

x y x

F x e dy e

E X 2Var X

1

1 X M t

t





If X follows Gamma distribution, find:

1. Probability Density Function2. Expected Value3. Variance4. Moment-Generating Function





1

, 0

0 , elsewhere

x xe x

f x

E X 2

Var X

1

1 X M t

t





If X follows normal distribution, find:

1. Probability Density Function2. Expected Value

3. Variance4. Moment-Generating Function





2

221 ,2

x

f x e x

E X 2Var X

2 2

2

t

M t e





If X follows normal distribution, find:







2

21

,2

x

f x e z

0 E X 1Var X





If1

n

i

i

S X

, then

1. E S 2. Var S





1

n

i

i

E S E X nE X

1

n

i

i

Var S Var X nVar X





If Y follows Lognormal distribution, find:







21 ln

21, 0

2

y

f y e y y

21

2 E Y e

2 22

1Var Y e e





Hazard Rate Function





f t t

S t





Weibull Distribution





1

, 0

x

x f y e x

1 , 0

x

F y e x

When 1 , the distribution becomes exponential.





Single-Parameter Pareto Distribution





1, f x x

x

1 ,F x x

x

1

E X





Two-Parameter Pareto Distribution





1

, 0 f x x

x

1F x

x

1

E X

Jensen Inequality and Chebychev’s Inequality




Jensen’s Inequality





( ) f x f x

, where 0 E h X hE X h X





Chebychev’s Inequality





2

1Pr X k

k

Application of Probability in Risk Management




Payment with Different Types of Deductible





For ordinary deductible, payment is made when

Payment max 0, X d where X is losses and d is the deductible

set.

For franchise deductible, payment is made when

Payment if X X d where X is losses and d is the deductible set.





Expected Payment with Deductibles


Under ordinary deductible,




Payment ( )d

E x d f x dx

Paymentd

E S x dx

Under franchise deductible,

Payment ( ) ( )d d

E xf x dx x d f x dx dS d





Expected Payment with Benefit Limit and Deductible


Under ordinary deductible




Under ordinary deductible,

0

Payment ( )

u

E x d f x dx u d S u

Under franchise deductible,

Payment ( ) ( )

u

d u

E xf x dx uf x dx





Relationship between the expected payment and the expectedpayment with deductible





^ E X E X d E X d whereby,

0 0^ ( ) ( )

d d

d E X d xf x dx d f x dx S x dx

( )d d

E X d x d f x dx S x dx





Coinsurance





Payment 1 1 E E X E X





Expected payment with deductible and coinsurance





Payment 1 ( )d

E x d f x dx





Expected payment with deductible, benefit limit and coinsurance





Payment 1 ( ) 1 ( )d d

E x d f x dx u d f x dx



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