The last form is the one that will be used in examinations.

Transcription

1 Tables for Exam M

2 The reading material for Exam M includes a variety of textbooks. Each text has a set of probability distributions that are used in its readings. For those distributions used in more than one text, the choices of parameterization may not be the same in all of the books. This may be of educational value while you study, but could add a layer of uncertainty in the examination. For this latter reason, we have adopted one set of parameterizations to be used in examinations. -TI- is set will be based on Appendices A & B of Loss Models: From Data to Decisions by Klugman, Pa~jer and Willmot. A slightly revised version of these appendices is included in this note. A copy of this note will also be distributed to each candidate at the examination. As an example of this adopted notation, consider the family of singleparameter exponential distributions. The distribution with mean 2 would be identified in three of the textbooks as follows: Actuarial Mathematics the exponential distribution with P = 1/2 Probability Models the exponential distribution with h = % Loss Models the exponential distribution with 0 =2 The last form is the one that will be used in examinations. Another difference among the texts is the choice of generating functions for discrete distributions. Loss Models uses probability generating functions while Actuarial Mathematics and Probability Models use moment generating functions. The abridged tables from Loss Models will provide only the probability generating 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 an examination question, the symbols will be defined. In addition to the abridged table from Loss Models, an abridged version of.the Illustrative Life Table from Actuarial Mathematics and a set of values from the standard normal distribution will be available for use in examinations. These are also included in this note.

3 NORMAL DISTRIBUTION TABLE Entries represent the area under the standardized normal distribution from -m to z, Pr(Zcz) The value of z to the first decimal is given in the left column. The second decimal place is given in the top row. Values of z for selected values of Pr(Z<z) z Pr(Z<z)

4 Excerpts from the Appendices to Loss Models: Rom Data to Decisions, 2nd edition May 27, 2005

5 Appendix A An Inventory of Continuous Distributions A.l Introduction The incomplete gamma function is given by Also, define with r(a) = t"-i e -t dt, a > 0. 1" 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 + Ic, a positive number. Then it can be evaluated from G(a + Ic; x) = r(a + Ic)[l - r(a + Ic; x)]. The incomplete beta function is given by

6 APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS A. 2 Transformed beta family A.2.3 A Three-parameter distributions Generalized Pareto (beta of the second kind)-a, 0, T + T) eaxr-l x F(x)=/~(T,~;u), a=- x+o r((~ = r(~)r(~) (X + e)*+r E[x~] = ekr(~+ k)r(a - k) r(a)r(l-) 1-7<k<a + E[xk] = ek7(7 1)... + (T k - 1), if Ic is an integer (a- 1). *.(a- k) ekr(7 + E[(X A x )~] k )r(~ = - k) P(7 + k, a - k; u) + xk[l - F(x)], k > -7 r(a)r(~) 7-1 mode = 0- T > 1, else 0 a+ll A Burr (Burr Type XII, Singh-Madda1a)-a, 0, y A Inverse Burr (Dagum)+, 0, y

7 APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS A.2.4 Two-parameter distributions A Pareto (Pareto Type 11, Lomax)--a, 0 E[xk] = ek k! (a-l)...(a-k)' if k is an integer mode = 0 A Inverse Pareto--r, 0 E[xk] = ek(-k)! (T- 1)...(~+ k)' if k is a negative integer 7-1 mode = et, r > 1, else 0 A Loglogistic (Fisk)--y, 0 mode = 0 (s) 117, y > 1, else 0

8 APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS A Paralogistic-a, 0 This is a Burr distribution with y = a. mode = 0 (s) liu", a>1, else0 A Inverse paralogistic-t, 0 This is an inverse Burr distribution with 7 = T. E[(XAX)~] = ekr(t + k'r)r(l - k'r)p(~ + k/r, 1 - k/r; u) + xk[l - ut], k > -r2 r(7) mode = ( ~ - l ) r>l, ~, else0 A.3 Transformed gamma family A.3.2 Two-parameter distributions A Gamma--a, 0 E[x~] = ek(a + k - 1)... a, if k is an integer - + 1)...(a + k - i)ekr(~ + k; ~/e) + xk[i - r (~; mode = e(a-1), a>1, else0 x/e)], k an integer

9 APPENDIX A. AN INVENTORY OF CONTINUO US DISTRIBUTIONS A Inverse gamma (Vinci)-a, 0 E[xk] - k), k<a r(a) E[x~] = ok (a-l)...(a- k)' if k is an integer mode + 1), mode = i 0 ( ), ~ > 1, else0 A Inverse Weibull (log Gompertz)-0, T E[(X Ax)*] k /~){l - r[l - klr; (@/x)~]} + xk [l- e-('/")'] all k mode = 0 (-&) 11' A.3.3 One-parameter distributions A Exponential-@ M(t) = (1-@t)-l ~[~*]=@*~(k+l), k> -1 E[x*] = okk!, if k is an integer E[X AX] e-"1') E[(X AX)" + l)r(k + 1; x/@) + xke-"le, k > -1 = ~~k!i?(k + 1; x/@) + xkee-pe, k an integer mode = 0

10 APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS A Inverse exponential-6 A.4 Other distributions E[xk] = ekr(i - k), k < 1 E[(X A x )~] = - k; e /~) + Xk(l - e-e/~ ) all k mode = el2 A Lognormal-p,o (p can be negative) 1 Ins p f(x) = - exp(-z2/2) = q5(z)/(ux), z = - xofi u E[xk] = exp(kp + k202/2) F(x) E[(X A x)*] = exp(kp + k2u2/2)@ ( lnx - p - ku2 u mode = exp(p - u2) A Inverse Gaussian-p, 0 f(x) = (L) 2~~~ '"exp (-g), z = - x P P FIX) = a[z(8)'12] +exp(t)a[-y(~)'12], ( t ) = exp [i ) (1 - :E[X A XI = x - p;@ [z (8) '1 - ] ) + xk[l - F(x)].=- P X+P e, t < -, E[X] = p, Var[X] = p3/e 2p2 py exp (T) a [-y (8) 'I2] A 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 the normal 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.,

11 APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS aek koa E[x~] = - k <a E[(XAx)]-a-k k (a-k)xa-k' x>e a-k' mode = 8 Note: Although there appears to be two parameters, only a is a true parameter. The value of 0 must be set in advance. A.5 Distributions with finite support aok For these two distributions, the scale parameter 0 is assumed known. A Generalized beta-a, b, 8, T E[(X A x )~] = + b)r(a k't)p(a + k/r, 6; u) + xk[l - ~ (a, b; u)] r(a)r(a + b + k /~) A beta-a, b, f (x) = -( r(a)r(b) x F(x) = P(a1 b;u) E[Xk] ekr(a + b)r(a + k) = > -a r(a)r(a+ b+ k) ' eka(a + 1)...(a + k - 1) E[x~] = o < x < o, (a+b)(a+b+l)-..(a+b+k-1)' u = ~ / e if k is an integer 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 b; u)]

12 Appendix B An Inventory of Discrete Distributions B.2 The (a,b,o) class B Poisson-X E[N] = B, Var[N] =B(l+B) P(z) = [l - p(z - I)]-l. This is a special case of the negative binomial with T = 1. B 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 Negative binomial-b, T

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

14 Illustrative Life Table: Basic Fun.ctions and Single Benefit Premiums at i =

15 Illustrative Life Table: Basic Functions and Sinale Benefit Premiums at i = 0.06 Lid96 are independent.

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

17 Interest Functions i i I lnterest Functions at i = 0.06 d"" idd o~d'ml Special Note: Unless specified, the force of interest is constant in each question. I