Title | SOA (P) formula sheet actuarial studies |
---|---|
Author | Savannah Gu |
Course | Actuarial Statistics |
Institution | University of New South Wales |
Pages | 4 |
File Size | 302 KB |
File Type | |
Total Downloads | 90 |
Total Views | 137 |
This is the formula sheet for SOA (P) society of Actuary...
Exam P updated 01/14/21
You have what it takes to pass GENERAL PROBABILITY
UNIVARIATE PROBABILITY DISTRIBUTIONS
BasicProbabilityRelationships Pr( ∪ ) = Pr() + Pr() − Pr( ∩ ) Pr( ∪ ∪ ) = Pr() + Pr() + Pr() − Pr( ∩ ) − Pr( ∩ ) − Pr( ∩ ) + Pr( ∩ ∩ ) Pr(! ) = 1 − Pr()
LawofTotalProbability Pr() = . Pr( ∩ " ) #
"
%$ DeMorgan’sLaw Pr[( ∪ )! ] = Pr(! ∩ ! ) Pr[( ∩ )! ] = Pr(! ∪ ! )
ConditionalProbability Pr( ∩ ) Pr(|) = Pr()
Independence Pr( ∩ ) = Pr() ⋅ Pr() Pr(|) = Pr()
Bayes’Theorem Pr(|& ) ⋅ Pr(& ) Pr(& |) = # ∑"$% Pr(|" ) ⋅ Pr(" )
Combinatorics ! = ⋅ ( − 1) ⋅ … ⋅ 2 ⋅ 1 ! #& = ( − )! ! #& = : ; = ( − )! ⋅ !
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*Learnbothdiscreteandcontinuouscases *ProbabilityMassFunction(PMF) ∑ ())+ ' ()= 1 Pr( = ) = 0(continuous)
*CumulativeDistributionFunction
(CDF) ' () = Pr( ≤ ) = ∑",+ ' () Pr( < ≤ ) = ' () − ' ()
' () =-+ ' ()(continuous) -
*ExpectedValue E[] = E[()] =
E[()] =
. ∫/. () ⋅ ' () . ! () , ∫0 ⋅ ' ()
for ≥ 0and(0) = 0
() ⋅ ' () E[()| ≤ ≤ ] = Pr( ≤ ≤ ) E[ ⋅ ()] = ⋅ E[()] E[% () + ⋯ + & ()] = E[% ()] + ⋯ + E[& ()] Variance,StandardDeviation,and CoefficientofVariation Var[] = E[2 ] − (E[])2 Var[ + ] = 2 ⋅ Var[] Var[] = 0 & 1∫
*MomentGeneratingFunction(MGF) ' () = E[ 3' ]
4'56 () = 63 ⋅ ' () ' (0) = 1 '57 () = ' () ⋅ 7 ()(independent) # ()[ = E[ # ] # ' 3$0
ProbabilityGeneratingFunction(PGF) ' () = E[' ] ' () = ' (ln ) ' (0) = ' (0) # ()[ # ' 3$0 = ' () ! # ()[ = E[( − 1) … ( − + 1)] # ' 3
%$ Percentiles The100thpercentileisthesmallestvalue
of8 where' _8 ` ≥ .
UnivariateTransformation 7 () = ' [/% ()] ⋅ [ /% ()[ where = () ⇔ = /% ()
SD[] = UVar[] CV[] = SD[] ⁄E[]
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1
DiscreteDistributions
1 , − + 1 = , + 1, … , : ; +( 1 − )#/+ , = 0, 1, … , − i , ;h : ; e: ; ⋅ : − = 0, 1, … ,
Mean
(1 − ); ,
1 − 1
PMF
DiscreteUniform
Binomial
Hypergeometric Geometric : trials; : failures = + 1
⋅
(1 − )+/% ,
1
= 1, 2, 3, …
PGF
−
SpecialProperties
–
–
(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: ( − | > )~ ( − | ≥ )~ NegBin( = 1, )~ Geometric()
Sumofindependent Poissons~Poisson ( =∑#"$% " )
ContinuousDistributions Continuous Uniform Exponential
Gamma
Normal
1 , − ≤ ≤ PDF
1 /+ A , > 0
+ B/% ⋅ /A , Γ() ⋅ B > 0
1 √2 −∞ < < ∞
(+/C )" / ⋅ 2D" ,
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CDF
− −
+
1 − . Pr( = ) , B/%
~Poisson: = ;, A = 1, 2, 3, … &$0
Variance
2
+ 2
1 − A /
Mean
+
− = Pr( ≤ ) = Φ()
( − )2 12
2
2
MGF
SpecialProperties
63 − 43 ( − )
(| > )~Uniform(, ) ( − | > )~Uniform(0, − )
B 1 x w 1 −
Sumofindependent exponentials()~ Gamma(, )
1 1 −
C352
D"3"
Memorylessproperty: ( − | > )~
Symmetry: Pr( ≤ ) = Pr ( ≥ −) Pr( ≤ −) = Pr( ≥ ) Sumofindependentnormals ~ Normal_ =∑ #"$% " , 2 =∑#"$% "2`
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2
MULTIVARIATE PROBABILITY DISTRIBUTIONS
*Learnbothdiscreteandcontinuouscases *JointPMFandCDF ∑())+ ∑()); ',7 (, )= 1 ',7 (, ) = ∑ F,+
∑3,; ',7 (,)
',7 (, ∞) = ' () ',7 (∞, ) = 7 () G"
G+G;
',7 (, ) = ',7 (, )(continuous)
*MarginalDistributionsand ConditionalDistributions ' () = ∑()); ',7 (, )
7 () = ∑())+ ',7 (, ) '|7 (| = ) = ',7 (, ) ⁄7 () *JointExpectedValueand ConditionalExpectation E[(, )]
= ∫/. ∫/. (, ) ⋅ ',7 (, ) .
.
E[| = ] =
. ∫ /.
⋅ '|7 (| = )
DoubleExpectationand LawofTotalVariance E[] = EáE[|]à
Var[] = EáVar[|]à + VaráE[|]à
CovarianceandCorrelationCoefficient 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
*JointMGF ',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
MultivariateTransformation J&,J" (% , 2 )
= '&,'" [ℎ% (% , 2 ), ℎ2 (% , 2 )] ⋅ || where % = ℎ% (% , 2 ) 2 = ℎ2 (% , 2 ) % % 2 = ê % 2 2ê % 2 MultinomialDistribution Pr(% = % , … , & = & ) ! = ⋅ +& ⋅ … ⋅ & +' % ! ⋅ … ⋅ & ! % E[" ] = " Var[" ] = " (1 − " ) Cová" , 1 à = −" 1 ,
≠
BivariateContinuousUniform 1 ',7 (, ) =Areaofdomain Areaofregion Pr(region) = Areaofdomain BivariateNormal For~Normal(' , ' 2 )and
~Normal(7 , 7 2 ), (| = )~Normal where
− ' E[| = ] = 7 + ⋅ 7 w x ' 2 2 Var[| = ] = 7 (1 − ) ExpectationandVarianceforSumand AverageofI.I.D.RandomVariables = % + ⋯ + # ò = [% + ⋯ + # ]⁄ E[] = ⋅ E[" ] E[ò] = E[" ] Var[] = ⋅ Var[" ] Var[ò] = (1/) ⋅ Var[" ]
CentralLimitTheorem Thesumoraverageofalargenumberof independentandidenticallydistributed (i.i.d.)randomvariablesapproximately followsanormaldistribution. OrderStatistics (%) = min(% , 2 , … , # ) (#) = max(% , 2 , … , # )
Fori.i.d.randomvariables, '(&) () = [' ()]# '(%) () = [' ()]#
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3
INSURANCE AND RISK MANAGEMENT
DefinitionofPayment,
Category
=ü
Deductible
PolicyLimit DeductibleandPolicyLimit (isthepolicylimit/ maximumpayment) UnreimbursedLoss,
0, − ,
= †
, ,
0, = ¢ − , ,
[]
≤ >
< ≥
≤ < < + ≥ +
. ∫- ( − ) ⋅ '( ) . ∫- '()
+ ⋅ ' () ∫0 ⋅ ' () K
∫-
K ∫0
' ()
( − ) ⋅ ' () + ⋅ ' ( + )
-5K
-5K
∫-
' ()
Forexponential: ⋅ Pr( > ) Forexponential: ⋅ Pr( < ) Forexponential: ⋅ Pr( < < + )
Ifisthelossandisthepayment(i.e.reimbursedloss),then = + ⇒ = − ,andE[] = E[] − E[].
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