A&J Flashcards for

Exam P/1 1st Edition

Alvin Soh

Outline

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A) Combinatorial Analysis

B) Basic Probability and 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 Variables with its Applications

I) Jensen Inequality and Chebyshev’s Inequality

J) Application of Probability in Risk Management

Basic Probabilities and Set Theory

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Distributive Law

Basic Probabilities and Set Theory

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A B C AC BC

AB C AC BC

Basic Probabilities and Set Theory

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De Morgan’s Law

Basic Probabilities and Set Theory

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1 1

Cn n

C

i i

i i

E E

1 1

Cn n

C

i i

i i

E E

Basic Probabilities and Set Theory

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Law of Total Probability

Basic Probabilities and Set Theory

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Pr Pr Pr PrA AB AC AD

n A n AB n AC n AD

Conditional Probability and Bayes Formula

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Bayes Formula

Conditional Probability and Bayes Formula

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Pr Pr | Pr

Pr |Pr Pr | Pr Pr | PrC C

AB B A AA B

B B A A B A A

Discrete Random Variables

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Characteristics of Probability Mass Function

Discrete Random Variables

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1. ( ) Pr( )f x X x

2. ( ) 0,f x x

3. ( ) 1x

f x

Discrete Random Variables

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thk moment

Discrete Random Variables

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: ( ) 0 : ( ) 0

( ) Prk k k

x f x x f x

E X x f x x X x

Discrete Probability Distribution

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If X follows binomial distribution, find:

1. Probability Mass Function 2. Expected Value 3. Variance

Discrete Probability Distribution

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The distribution is commonly represented by

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

nX x p p

x

, where

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

E X np 1Var X np p

Continuous Random Variable

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Transformation of Random Variables

using

“CDF” Method

Continuous Random Variable

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

depending on the function, g

Continuous Random Variable

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Transformation of Random Variables

using

“Inverse Transformation” Method

Continuous Random Variable

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

'

Y Xf y f h y h y , where 1h y g y

Continuous Probability Distribution

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If X follows Exponential distribution, find:

1. Probability Density Function 2. Expected Value 3. Variance 4. Moment-Generating Function

Continuous Probability Distribution

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1

, 0

0 , elsewhere

x

e xf x

0

11

x y x

F x e dy e

E X 2Var X

1

1XM t

t

Application of Probability in Risk Management

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Relationship between the expected payment and the expected payment with deductible

Application of Probability in Risk Management

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^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

Application of Probability in Risk Management

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Coinsurance

Application of Probability in Risk Management

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Payment 1 1E E X E X

Application of Probability in Risk Management

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