P formula sheet sheet sheet sheet PDF

Preview

Summary

P formula sheet sheet sheet sheetP formula sheet sheet sheet sheetP formula sheet sheet sheet sheet...

Description

Exam P updated 01/14/21

You have what it takes to pass GENERAL PROBABILITY

UNIVARIATE PROBABILITY DISTRIBUTIONS



BasicProbabilityRelationships Pr(฀฀ ∪ ฀฀) = Pr(฀฀) + Pr(฀฀) − Pr(฀฀ ∩ ฀฀) Pr(฀฀ ∪ ฀฀ ∪ ฀฀ ) = Pr(฀฀) + Pr(฀฀) + Pr(฀฀) − Pr(฀฀ ∩ ฀฀) − Pr(฀฀ ∩ ฀฀) − Pr(฀฀ ∩ ฀฀) + Pr(฀฀ ∩ ฀฀ ∩ ฀฀) Pr(฀฀! ) = 1 − Pr(฀฀)

 LawofTotalProbability Pr(฀฀) = . Pr(฀฀ ∩ ฀฀" ) #

"

%$ DeMorgan’sLaw Pr[(฀฀ ∪ ฀฀)! ] = Pr(฀฀! ∩ ฀฀ ! ) Pr[(฀฀ ∩ ฀฀)! ] = Pr(฀฀! ∪ ฀฀ ! )

 ConditionalProbability Pr(฀฀ ∩ ฀฀) Pr(฀฀|฀฀) =  Pr(฀฀)

 Independence Pr(฀฀ ∩ ฀฀) = Pr(฀฀) ⋅ Pr(฀฀) Pr(฀฀|฀฀) = Pr(฀฀)

 Bayes’Theorem Pr(฀฀|฀฀& ) ⋅ Pr(฀฀& ) Pr(฀฀& |฀฀) = #  ∑"$% Pr(฀฀|฀฀" ) ⋅ Pr(฀฀" )

 Combinatorics ฀฀! = ฀฀ ⋅ (฀฀ − 1) ⋅ … ⋅ 2 ⋅ 1 ฀฀! #฀฀& = (฀฀ − ฀฀)! ฀฀! ฀฀ #฀฀& = : ฀฀ ; = (฀฀ − ฀฀)! ⋅ ฀฀!

www coachingactuaries com



*Learnbothdiscreteandcontinuouscases  *ProbabilityMassFunction(PMF) ∑ ())+ ฀฀' (฀฀)= 1 Pr(฀฀ = ฀฀) = 0(continuous)

 *CumulativeDistributionFunction

(CDF) ฀฀' (฀฀) = Pr(฀฀ ≤ ฀฀) = ∑",+ ฀฀' (฀฀) Pr(฀฀ < ฀฀ ≤ ฀฀) = ฀฀' (฀฀) − ฀฀' (฀฀)

฀฀' (฀฀) =-+ ฀฀' (฀฀)(continuous) -

 *ExpectedValue E[฀฀] = ฀฀ E[฀฀(฀฀)] =

E[฀฀(฀฀)] =

.  ∫/. ฀฀(฀฀) ⋅ ฀฀' (฀฀)฀฀฀฀ . ! ฀฀ (฀฀) , ∫0 ⋅ ฀฀' (฀฀)฀฀฀฀

for฀฀ ≥ 0and฀฀(0) = 0

฀฀(฀฀) ⋅ ฀฀' (฀฀)฀฀฀฀  E[฀฀(฀฀)|฀฀ ≤ ฀฀ ≤ ฀฀] = Pr(฀฀ ≤ ฀฀ ≤ ฀฀) E[฀฀ ⋅ ฀฀(฀฀)] = ฀฀ ⋅ E[฀฀(฀฀)] E[฀฀% (฀฀) + ⋯ + ฀฀& (฀฀)] = E[฀฀% (฀฀)] + ⋯ + E[฀฀& (฀฀)]  Variance,StandardDeviation,and CoefficientofVariation Var[฀฀] = E[฀฀2 ] − (E[฀฀])2   Var[฀฀฀฀ + ฀฀] = ฀฀2 ⋅ Var[฀฀]  Var[฀฀] = 0 & 1∫

*MomentGeneratingFunction(MGF) ฀฀' (฀฀) = E[฀฀ 3' ]

฀฀4'56 (฀฀) = ฀฀ 63 ⋅ ฀฀' (฀฀฀฀) ฀฀' (0) = 1 ฀฀'57 (฀฀) = ฀฀' (฀฀) ⋅ ฀฀7 (฀฀)(independent) ฀฀# ฀฀ (฀฀)[ = E[฀฀ # ] ฀฀฀฀ # ' 3$0

 ProbabilityGeneratingFunction(PGF) ฀฀' (฀฀) = E[฀฀' ] ฀฀' (฀฀) = ฀฀' (ln ฀฀) ฀฀' (0) = ฀฀' (0) ฀฀# ฀฀ (฀฀)[ ฀฀฀฀ # ' 3$0 = ฀฀' (฀฀) ฀฀! # ฀฀ ฀฀ (฀฀)[ = E[฀฀(฀฀ − 1) … (฀฀ − ฀฀ + 1)] ฀฀฀฀ # ' 3

%$ Percentiles The100฀฀thpercentileisthesmallestvalue

of฀฀8 where฀฀' _฀฀8 ` ≥ ฀฀.

 UnivariateTransformation ฀฀ ฀฀7 (฀฀) = ฀฀' [฀฀/% (฀฀)] ⋅ [ ฀฀/% (฀฀)[ ฀฀฀฀ where฀฀ = ฀฀(฀฀) ⇔ ฀฀ = ฀฀/% (฀฀)

SD[฀฀] = UVar[฀฀] CV[฀฀] = SD[฀฀] ⁄E[฀฀]



ht © 2021 Coaching Actuaries All Rights Reserved

1



 DiscreteDistributions  

1 , ฀฀ − ฀฀ + 1 ฀฀ = ฀฀, ฀฀ + 1, … , ฀฀ ฀฀ : ; ฀฀+( 1 − ฀฀)#/+ ,  ฀฀ ฀฀ = 0, 1, … , ฀฀ ฀฀ ฀฀ − ฀฀ i ฀฀ , ;h : ; e: ฀ ฀ ; ⋅ : ฀฀ ฀฀ − ฀฀ ฀฀ = 0, 1, … , ฀฀

Mean

(1 − ฀฀); ฀฀,

1 − 1 ฀฀

PMF

DiscreteUniform

Binomial

Hypergeometric Geometric  ฀฀: trials; ฀฀: failures ฀฀ = ฀฀ + 1

฀฀฀฀

฀฀ ⋅

(1 − ฀฀)+/% ฀฀,

฀฀  ฀฀

1  ฀฀

฀฀ = 1, 2, 3, …

PGF

฀฀ − ฀฀ ฀฀ ฀฀

SpecialProperties

–

–

฀฀฀฀(1 − ฀฀) 

(฀฀฀฀ 3 + ฀฀ )# 

( ฀฀฀฀ + ฀฀ )# 

–

–

–

–

–

1 − ฀฀  ฀฀2

฀฀  ฀฀

฀฀ /= ⋅ ฀฀+ , ฀฀! ฀฀ = 0, 1, 2, …

MGF

(฀฀ − ฀฀ + 1)2 − 1 ฀฀ 43 − ฀฀ (65%)3  12 (1 − ฀฀3 )(฀฀ − ฀฀ + 1)

฀฀ + ฀฀  2

฀฀ = 0, 1, 2, … ฀฀ − 1 < w x ฀฀ (1 − ฀฀)+/< , Negative ฀฀ − 1 ฀฀ = ฀฀, ฀฀ + 1, 2, … Binomial ฀฀: trials; ฀฀: failures ฀฀ + ฀฀ − 1 < w x ฀฀ (1 − ฀฀); , ฀฀ − 1 ฀฀ = ฀฀ + ฀฀ ฀฀ = 0, 1, 2, … Poisson

Variance

1 − ฀฀ ฀฀ w 2 x ฀฀

฀฀฀฀  1 − (1 − ฀฀ )฀฀

฀฀฀฀ 3  1 − (1 − ฀฀ )฀฀3

฀฀  1 − (1 − ฀฀ )฀฀3

฀฀฀฀ 3 y z 1 − (1 − ฀฀)฀฀3

<

< ฀฀ w x 1 − (1 − ฀฀)฀฀3

฀฀

฀฀ =>?

!/%@

฀฀  1 − (1 − ฀฀ )฀฀

< ฀฀฀฀ w x 1 − (1 − ฀฀)฀฀ < ฀฀ w x 1 − (1 − ฀฀)฀฀

฀฀ =(3/%)



Memoryless property: (฀฀ − ฀฀|฀฀ > ฀฀)~฀฀ (฀฀ − ฀฀ |฀฀ ≥ ฀฀ )~฀฀ NegBin(฀฀ = 1, ฀฀)~ Geometric(฀฀)

Sumofindependent Poissons~Poisson (฀฀ =∑#"$% ฀฀" )

ContinuousDistributions  Continuous Uniform Exponential

Gamma

Normal

1 , ฀฀ − ฀฀ ฀฀ ≤ ฀฀ ≤ ฀฀ PDF

1 /+ ฀฀A , ฀฀ ฀฀ > 0

+ ฀฀ B/% ⋅ ฀฀/A , Γ(฀฀) ⋅ ฀฀ B ฀฀ > 0

1 ฀฀√2฀฀ −∞ < ฀฀ < ∞

(+/C )" / ⋅ ฀฀ 2D" ,

www coachingactuaries com

CDF

฀฀ − ฀฀  ฀฀ − ฀฀

+

1 − . Pr(฀฀ = ฀฀) , B/%

฀฀~Poisson:฀฀ = ;, A ฀฀ = 1, 2, 3, … &$0

Variance

฀฀

฀฀ 2 

฀฀ + ฀฀  2

1 − ฀฀ A  /

Mean

+

฀฀ − ฀฀ ฀฀ =  ฀ ฀  Pr(฀฀ ≤ ฀฀) = Φ(฀฀)

฀฀฀฀

฀฀

(฀฀ − ฀฀)2  12

฀฀฀฀ 2 

฀฀ 2 

MGF

SpecialProperties

฀฀ 63 − ฀฀ 43  ฀฀(฀฀ − ฀฀)

(฀฀|฀฀ > ฀฀)~Uniform(฀฀, ฀฀) (฀฀ − ฀฀|฀฀ > ฀฀)~Uniform(0, ฀฀ − ฀฀)

B 1 x  w 1 − ฀฀฀฀

Sumof฀฀independent exponentials(฀฀)~ Gamma(฀฀, ฀฀)

1  1 − ฀฀฀฀

฀฀ C352 

D"3"

Memorylessproperty: ( ฀฀ − ฀฀|฀฀ > ฀฀)~฀฀

Symmetry: Pr(฀฀ ≤ ฀฀) = Pr (฀฀ ≥ −฀฀) Pr(฀฀ ≤ −฀฀) = Pr(฀฀ ≥ ฀฀)  Sumofindependentnormals ~ Normal_฀฀ =∑ #"$% ฀฀" , ฀฀ 2 =∑#"$% ฀฀"2`

Copyright © 2021 Coaching Actuaries All Rights Reserved

2

MULTIVARIATE PROBABILITY DISTRIBUTIONS 

*Learnbothdiscreteandcontinuouscases *JointPMFandCDF ∑())+ ∑()); ฀฀',7 (฀฀, ฀฀ )= 1 ฀฀',7 (฀฀, ฀฀) = ∑ F,+

∑3,; ฀฀',7 (฀฀,฀฀)

฀฀',7 (฀฀, ∞) = ฀฀' (฀฀) ฀฀',7 (∞, ฀฀) = ฀฀7 (฀฀) G"

G+G;

฀฀',7 (฀฀, ฀฀) = ฀฀',7 (฀฀, ฀฀ )(continuous)

 *MarginalDistributionsand ConditionalDistributions ฀฀' (฀฀) = ∑()); ฀฀',7 (฀฀, ฀฀) 

฀฀7 (฀฀) = ∑())+ ฀฀',7 (฀฀, ฀฀)  ฀฀'|7 (฀฀|฀฀ = ฀฀) = ฀฀',7 (฀฀, ฀฀) ⁄฀฀7 (฀฀)  *JointExpectedValueand ConditionalExpectation E[฀฀(฀฀, ฀฀)]

= ∫/. ∫/. ฀฀(฀฀, ฀฀) ⋅ ฀฀',7 (฀฀, ฀฀)฀฀฀฀ ฀฀฀฀ .

.

E[฀฀|฀฀ = ฀฀] =

. ∫ /.

฀฀ ⋅ ฀฀'|7 (฀฀|฀฀ = ฀฀ )฀฀฀฀ 

 DoubleExpectationand LawofTotalVariance E[฀฀] = EáE[฀฀|฀฀]à

Var[฀฀] = EáVar[฀฀|฀฀]à + VaráE[฀฀|฀฀]à

 CovarianceandCorrelationCoefficient Cov[฀฀, ฀฀] = E[฀฀฀฀] − E[฀฀]E[฀฀] Cov[฀฀฀฀, ฀฀฀฀] = ฀฀฀฀ ⋅ Cov[฀฀, ฀฀] Cov[฀฀, ฀฀] = Var[฀฀] Var[฀฀฀฀ + ฀฀฀฀] = ฀฀ 2 Var[฀฀] + ฀฀ 2 Var[฀฀]  + 2฀฀฀฀ ⋅ Cov[฀฀, ฀฀ ] 

฀฀',7 = Corr[฀฀, ฀฀] =

Cov[฀฀, ฀฀]

 UVar[฀฀]UVar[฀฀]

Independence

฀฀',7 (฀฀, ฀฀) = ฀฀' (฀฀) ⋅ ฀฀7 (฀฀) ฀฀',7 (฀฀, ฀฀) = ฀฀' (฀฀) ⋅ ฀฀7 (฀฀) E[ℎ(฀฀) ⋅ ฀฀ (฀฀)] = E[ℎ(฀฀)] ⋅ E[฀฀(฀฀)] ฀฀',7 (฀฀, ฀฀) = ฀฀' (฀฀) ⋅ ฀฀7 (฀฀) Cov[฀฀, ฀฀] = 0 ฀฀',7 = 0

 *JointMGF ฀฀',7 (฀฀, ฀฀) = E[฀฀ F'537 ]

E[฀฀] =

E[฀฀] =

G

GF G

฀฀',7 (฀฀, ฀฀)ç

฀฀',7 (฀฀, ฀฀)ç

F$3$0

 

E[฀฀ I ฀฀# ] = GF#G3 % ฀฀',7 (฀฀, ฀฀)ç G3

F$3$0

G#

%$฀฀',7 (฀฀, ฀฀) = ฀฀'57 (฀฀)

F$3$0

 MultivariateTransformation ฀฀J&,J" (฀฀% , ฀฀2 )



= ฀฀'&,'" [ℎ% (฀฀% , ฀฀2 ), ℎ2 (฀฀% , ฀฀2 )] ⋅ |฀฀| where ฀฀% = ℎ% (฀฀% , ฀฀2 ) ฀฀2 = ℎ2 (฀฀% , ฀฀2 ) ฀฀฀฀% ฀฀฀฀% ฀฀฀฀ ฀฀฀฀2 ฀฀ = ê %  ฀฀฀฀2 ฀฀฀฀2ê ฀฀฀฀% ฀฀฀฀2  MultinomialDistribution Pr(฀฀% = ฀฀% , … , ฀฀& = ฀฀& ) ฀฀! = ⋅ ฀฀ +& ⋅ … ⋅ ฀฀& +'  ฀฀% ! ⋅ … ⋅ ฀฀& ! % E[฀฀" ] = ฀฀฀฀"  Var[฀฀" ] = ฀฀฀฀" (1 − ฀฀" ) Cová฀฀" , ฀฀1 à = −฀฀฀฀" ฀฀1 ,

฀฀ ≠ ฀฀

BivariateContinuousUniform  1 ฀฀',7 (฀฀, ฀฀) =Areaofdomain Areaofregion  Pr(region) = Areaofdomain  BivariateNormal For฀฀~Normal(฀฀' , ฀฀' 2 )and

฀฀~Normal(฀฀7 , ฀฀7 2 ),  (฀฀|฀฀ = ฀฀ )~Normal where

฀฀ − ฀฀' E[฀฀|฀฀ = ฀฀ ] = ฀฀7 + ฀฀ ⋅ ฀฀7 w x ฀฀' 2 2 Var[฀฀|฀฀ = ฀฀] = ฀฀7 (1 − ฀฀ )  ExpectationandVarianceforSumand AverageofI.I.D.RandomVariables ฀฀ = ฀฀% + ⋯ + ฀฀#  ฀฀ò = [฀฀% + ⋯ + ฀฀# ]⁄ ฀฀  E[฀฀] = ฀฀ ⋅ E[฀฀" ] E[฀฀ò] = E[฀฀" ] Var[฀฀] = ฀฀ ⋅ Var[฀฀" ] Var[฀฀ò] = (1/฀฀) ⋅ Var[฀฀" ] 

CentralLimitTheorem Thesumoraverageofalargenumberof independentandidenticallydistributed (i.i.d.)randomvariablesapproximately followsanormaldistribution.  OrderStatistics ฀฀(%) = min(฀฀% , ฀฀2 , … , ฀฀# ) ฀฀(#) = max(฀฀% , ฀฀2 , … , ฀฀# ) 

Fori.i.d.randomvariables, ฀฀'(&) (฀฀) = [฀฀' (฀฀)]#  ฀฀'(%) (฀฀) = [฀฀' (฀฀)]#  



www coachingactuaries com

Copyright © 2021 Coaching Actuaries All Rights Reserved

3

INSURANCE AND RISK MANAGEMENT 

DefinitionofPayment, ฀฀

Category

฀฀ =ü

Deductible

PolicyLimit DeductibleandPolicyLimit (฀฀isthepolicylimit/ maximumpayment)  UnreimbursedLoss,฀฀

0, ฀฀ − ฀฀,

฀฀ = †

฀฀, ฀฀,

0, ฀฀ = ¢฀฀ − ฀฀, ฀฀,

฀฀[฀฀]

฀฀ ≤ ฀฀  ฀฀ > ฀฀

฀฀ < ฀฀  ฀฀ ≥ ฀฀

฀฀ ≤ ฀฀ ฀฀ < ฀฀ < ฀฀ + ฀฀ ฀฀ ≥ ฀฀ + ฀฀

. ∫- (฀฀ − ฀฀) ⋅ ฀฀'( ฀฀) ฀฀฀฀ . ∫- ฀฀'(฀฀)฀฀฀฀

+ ฀฀ ⋅ ฀฀' (฀฀) ∫0 ฀฀ ⋅ ฀฀' (฀฀)฀฀฀฀ K

∫-

K ∫0

฀฀' (฀฀)฀฀฀฀ 

(฀฀ − ฀฀) ⋅ ฀฀' (฀฀)฀฀฀฀ + ฀฀ ⋅ ฀฀' (฀฀ + ฀฀ )

-5K

-5K

∫-

฀฀' (฀฀)฀฀฀฀ 

Forexponential: ฀฀ ⋅ Pr( ฀฀ > ฀฀) Forexponential: ฀฀ ⋅ Pr(฀฀ < ฀฀) Forexponential: ฀฀ ⋅ Pr(฀฀ < ฀฀ < ฀฀ + ฀฀ )

If฀฀isthelossand฀฀isthepayment(i.e.reimbursedloss),then฀฀ = ฀฀ + ฀฀ ⇒ ฀฀ = ฀฀ − ฀฀,andE[฀฀] = E[฀฀] − E[฀฀].

www coachingactuaries com

Copyright © 2021 Coaching Actuaries. All Rights Reserved. 4 Personal copies permitted Resale or distribution is prohibited...