A&J PFlashcards JosephShackelford

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A&J Flashcards for

Exam P/11st Edition

Alvin Soh, ASA, CERA

Outline

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 =

! ! !r

Multinomial Coefficient

The number of division of objects into distinctive groups of size

is given by:

1 2, , r n n n

1 2 1 2

, , , ! ! !r r

n n n n n n

Basic Probabilities and Set Theory

Commutative Law

A B B A

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

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 n B n AB

Pr Pr Pr Pr A B A B AB

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

A B C AB AC BC ABC

Definition of Probability

# of outcomes in the event

Probability that event A happens = PrTotal # of possible event

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 Pr Pr AB A B

Conditional Probability and Bayes Formula

Pr | A B

PrPr |

n AB AB A B

(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

: ( ) 0 : ( ) 0

( ) Pr x f x x f x

E X xf x x X x

( ) 1 ( ) , x n

E X S x F x x

Di R d V i bl

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 .

: ( ) 0

( ) ( ) x f x

E g X g x f x

Moment-Generating Functions

M t E e e f x

Relationship between thk moment and moment-generating

function

d M t E X

Variance, Var X

Var X E X E X

Var aX b

2Var aX b a Var X

Standard Deviation, 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.

a b E X

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

, 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!

where is the parameter for this distribution.

E X Var X

If X follows Geometric distribution, find:

~ X Geo p , with 1

where p is the parameter for this distribution.

1Var X

If X follows Negative Binomial distribution, find:

~ , X NB r p , with 1

X x p pr

where r and p are the parameters for this distribution.

1r pVar X

If X follows Hypergeometric distribution, find:

~ , , X HyperGeo N n r , with Pr

n x x X x

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

E X n N

r r N nVar X n

Continuous Random Variable

Characteristics of Continuous Random Variables

1. Pr 0 X x f x

a X b f x dx

3. 1 f x dx

Expected Value and Variance

E X xf x dx

E X S x dx

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

“CDF” Method

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

depending on the function, g

Transformation of Random Variables

“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

, a x bf

0 , elsewhere

f x b a

x aF x a x b

a b E X 2

b aVar X

If X follows Beta distribution, find:

11 1 , 0 1

0 , elsewhere

x x x f x

1Var X

If X follows Exponential distribution, find:

2. Expected Value3. Variance4. Moment-Generating Function

0 , elsewhere

e x f x

F x e dy e

E X 2Var X

1 X M t

If X follows Gamma distribution, find:

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

0 , elsewhere

x xe x

1 X M t

If X follows normal distribution, find:

1. Probability Density Function2. Expected Value

3. Variance4. Moment-Generating Function

221 ,2

f x e x

E X 2Var X

If X follows normal distribution, find:

f x e z

0 E X 1Var X

, then

1. E S 2. Var S

E S E X nE X

Var S Var X nVar X

If Y follows Lognormal distribution, find:

f y e y y

2 E Y e

1Var Y e e

Hazard Rate Function

Weibull Distribution

x f y e x

F y e x

When 1 , the distribution becomes exponential.

Single-Parameter Pareto Distribution

1, f x x

1 ,F x x

Two-Parameter Pareto Distribution

, 0 f x 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

1Pr X 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

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,

Payment ( )

E x d f x dx u d S u

Under franchise deductible,

Payment ( ) ( )

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 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

A&J PFlashcards JosephShackelford

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