Exam M Tables

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Tables forExam M


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The reading material for Exam M includes a variety of textbooks. Eachtext has a set of probability distributions that are used in its readings . For thosedistributions used in m ore than on e text, the choices of parameterization m ay notbe the same in all of the boo ks. This may b e of educational value while youstudy, but could add a layer of uncertainty in the exa mination. For this latterreason, we have adopted one set of parameterizations to be used inexaminations. -TI- is set will be based on Appendices A & B of Loss Models:From Data to Decisions by Klugman, P a ~ j e rnd Willmot. A slightly revisedversion of these appendices is included in this note. A copy of this note will alsobe distributed to each candidate at the exam ination.

As an e xample of this adopted notation, consider the family of single-parameter exponential distributions. The distribution with me an 2 would beidentified in three of the textbooks as follows:Actuarial Mathem atics the expone ntial distribution withP = 1/2Probab ility Mod els the expone ntial distribution withh = %Loss Models the expon ential distribution with0 =2The last form is the one that will be used in examinations.

Another difference among the texts is the choice of generating functionsfor discrete distributions. Loss Models uses probability generating functionswhile Actuarial Mathematics and Probability Models use moment generatingfunctions. The abridged tables from Loss Models will provide only the probabilitygenerating function for discrete distributions.Each text also has its own system of dedicated notation and terminology.Sometimes these may conflict. If alternative meanings could apply in anexamination question, the symbols will be defined.In addition to the abridged table from Loss Models, an abridged version of.the Illustrative L ife Table from Actuarial Mathematics and a set of values from thestandard normal distribution will be available for use in examinations. These arealso included in this note.


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NORMAL DISTRIBUTION TABLEEntries represent the area under the standardized normal distribution from -m to z, Pr(Zcz)The value of z to the first decim al is given in the left column . The second decim al pla ce is given in the top row.

Values of z for selected values of Pr(Z


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Excerpts from the Appendices to Loss Models: R o m Data toDecisions, 2nd edition

May 27, 2005


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Appendix AAn Inventory of ContinuousDistributionsA . l IntroductionThe incomplete gamma function is given by

with r ( a )= t"-I -t1" e dt, a > 0.Also, define

At times we will need this integral for nonpositive values of a. Integration by parts produces the relationship

This can be repeated until the first argument of G is a + c, a positive number. Then it can be evaluatedfrom

G(a+ Ic; x) = r ( a+ Ic)[l- ( a + Ic; x)].The incomplete beta function is given by


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APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONSA.2 Transformed beta familyA.2.3 Three-parameter distributionsA.2.3.1 Generalized Pareto (beta of th e second kind)-a, 0,T

r ( ( ~T) eaxr-l x= r ( ~ ) r ( ~ )X + e)*+r F ( x ) = / ~ ( T , ~ ; u ) ,= -x + o

E [ x ~ ] = e k r ( ~ +)r(a- )r(a)r(l-) 1 - 7 < k < aE [ xk ] = ek7 ( 7+ 1). . T + k - 1), if Ic is an integer( a - 1). .(a- k)

E[(XA x ) ~ ]= ekr(7 k ) r ( ~ )r ( a ) r ( ~ ) P(7+ k , a - k; u) + xk[l- F(x)], k > -77 - 1mode = 0 T > 1, else 0a + l lA.2.3.2 Burr (Burr Type XII, Singh-Madda1a)-a, 0,y

A.2.3.3 Inverse Bur r (Dagum)+, 0 ,y


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APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONSA.2.4 Two-parameter distributionsA.2.4.1 Pareto (Pareto Type 11, Lomax)--a, 0

E [ x k ] = ekk! if k is an integer( a - l ) . . . ( a - k ) '

mode = 0A.2.4.2 Inverse Pareto--r, 0

e k ( -k ) !E [ x k ] = if k is a negative integer( T - 1 ) . . . ( ~ + )'

7 - 1mode = e T , r > 1 , else 0A.2.4.3 Loglogistic (Fis k)-- y,0

117mode = 0(s)y > 1, else 0


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APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS

A.2.4.4 P a r a l o g i s t i c - a , 0This is a Burr distribution with y = a .

mode = 0(s)iu ", a > 1 , else0A.2.4.5 I n v e r s e p a r a l o gi s t ic - T , 0

This is an inverse Burr distribution with 7=T .

E [ ( X A X ) ~ ] ek r (T +k'r)r(l - k ' r ) p ( ~+ k / r , 1 - k / r ;u ) + x k [ l- u T ] , k > -r2r(7)mode = ( ~ - l ) ~ ,> l , else0

A.3 Transformed gamma familyA.3.2 Two-parameter distributionsA.3.2.1 Gamma--a, 0

E [ x ~ ] ek(a+ k - 1 ) . .a , if k is an integer

- + 1 ) . .( a+ k - i ) e k r ( ~k ;~ / e ) x k [ i- r ( ~ ;/ e ) ] , k an integermode = e ( a - 1 ) , a > 1 , else0


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APPENDIX A. AN INVENTORY O F CONTINUOUS DISTRIBUTIONS

A.3.2.2 I n v e r s e g a m m a ( V i n c i ) - a , 0

E [ x k ] = @"(a - k ) , k < a E [ x ~ ] okr ( a )

if k is an integer( a - l ) . . . ( a - k ) '

mode = @ / ( a+ 1 )

,-- 1 117mode = 0 ( , ~ > 1 ,lse0A.3.2.4 I n v e r s e W e i b u l l ( l o g G o m p e r t z ) - 0 , T

E [ ( X A x ) * ] = @ * r ( l k / ~ ) { l r[ l- k l r ; @ / x ) ~ ] }x k [ l- -( '/") '] all k11mode = 0 (-&)

A.3.3 One-parameter distributionsA.3.3.1 Ex pone n t ia l - @

M ( t ) = ( 1 - @ t ) - l ~ [ ~ * ] = @ * ~ ( k + l ) ,> -1E[x *] = o k k ! , if k is an integer

E [ X A X ] = @ ( I - e-"1')E [ ( X AX)" = @ * r ( k+ l ) r ( k+ 1;x / @ )+ x k e - " l e , k > -1

= ~ ~ k ! I ? ( k1;x / @ )+ x k e e - P e , k an integermode = 0


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APPENDIX A. AN INVENTORY O F CONTINUOUS DISTRIBUTIONSA.3.3.2 Inverse exponential-6

E[xk] = ekr(i k), k < 1E[(XA x ) ~ ]= - k; e / ~ ) Xk(l - e- e/ ~ all kmode = el2

A.4 Other distributionsA.4.1.1 Lognormal-p,o (p can b e negative)

1 Ins - pf(x ) =- xp(-z2/2) = q5(z)/(ux), z =-o f i F(x) = @(z)uE [x k ] = exp(kp+k202/2)

E[(X A x)*] = exp(kp+ k2u2/2)@ nx - p - ku2u ) + x k [ l- F(x)]mode = exp(p- u2)

A.4.1.2 Inve rse Gaussian-p, 0

x - Pf(x) = (L)~~~ '"exp (-g) z =-X + PFIX) = a[z(8)'12] e x p ( T ) a [ - y ( ~ ) ' 1 2 ] , .=- P

e( t ) = exp[i1 -) ] t < -, E[X]= p, Var[X]= p3/e2p2:E[X XI = x - p;@ [z (8) '1 - py exp (T) a [-y (8) 'I2]A.4.1.3 log-t-r, p, u ( p can be negative)

Let Y have a t distribution with r degrees of freedom. Then X = exp(uY + p) has the log-t distribution.Positive moments do not exist for this distribution. Just as the t distribution has a heavier tail than thenormal distribution, this distribution has a heavier tail than the lognormal distribution.

F(x) = F, (%) with F,(t) the cdf of a t distribution with 7 d.f.,


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APPENDIX A. AN INVENTORY O F CONTINUOUS DISTRIBUTIONS

aekE [ x ~ ]= - < a E [ ( X A x ) ] - a - k (a-k)xa-k' x > ea o kk koaa - k 'mode = 8

Note: Although there appears to be two parameters, only a is a true parameter. The value of 0 must beset in advance.

A.5 Distributions with finite supportFor these two distributions, the scale parameter 0 is assumed known.

A.5.1.1 Generalized b e t a - a , b, 8,T

E[(X A x ) ~ ]= + b)r(a + k'T)p(a + k/ r, 6; u) +xk[ l- ~ ( a ,; u)]r ( a ) r ( a+ b + k / ~ )A.5.1.2 beta-a, b, 8

1+ ( 1 - - < x < o, u =~ / ef (x) =-(a)r(b) xF(x) = P(a1 b;u)

E[Xk] = ekr (a+ b)r(a+ k) > -ar ( a ) r ( a + b + k) 'eka(a+ 1) . . . a + k - 1)E [ x ~ ] = if k is an integer( a + b ) ( a + b + l ) - . . ( a + b + k - 1 ) 'eka(a+ 1) . .(a +k - 1)E[(X A x ) ~ ]= ( a + b ) ( a + b + l ) . . . ( a + b + k - 1 ) P(a + k1 b; 21)

+xk [I- @(a,; u)]


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Appendix BAn Inventory of DiscreteDistributionsB.2 The ( a , b ,O ) classB.2.1.1 Poisson-X

E[N] = B, Var[N] = B ( l + B ) P(z)= [l- p( z - ) ] - l .This is a special case of the negative binomial with T = 1.

B.2.1.3 Binomial-q, m, (0 < q < 1, m an integer)

pr = (:)q*(l -q)m-k, t = 0 , 1 , . . ,m:E[N] = mq, Var[N]=mq(1- q) P(z)= [I+ q(z - I)]".

B.2 .1. 4 Negative binomial-B, T


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Illustrative Life Table: Basic Functions and Single Benefit Premiums at i =0.06


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Illustrative Life Table: Basic Fun.ctionsand Single Benefit Premiums at i = 0.06


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IllustrativeLife Table: Basic Functions and Sinale Benefit Premiumsat i = 0.06Lid96are ndependent.


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Illustrative Life Table: Basic Functions and Sinale Benefit Premiums at i = 0.06Livesare independent.


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Interest Functions

lnterest Functions at i = 0.06d"" idd o~d 'ml

Special Note: Unless specified, the force of interest is constant in each question .

Exam M Tables

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