Formulae and tables for actuarial exams

138 Pages·2010·5.03 MB·English

FORMULAE AND TABLES
FOR
ACTUARIAL .EXAM I NATIONS
(INSTITUTE OF ACTUARIES)
THE INSTITUTE OF ACTUARIES
AND
THE FACULTY OF ACTUARIES
1980
First published 19x0
, ' Reprinted 1983
Reprinted 19XX
Rcpriiited 1990
* I R eprinted 199 I
Pi-inled in Grar Drit,iin
:it the 4lden Press. Oul'ord
FORMULAE AND TABLES
FOR
ACTUARIAL EXAMINATIONS
(INSTITUTE OF. ACTUARIES)
INSTITUTE OF ACTUARIES
Formulae for the use of Candidates at the Examinations
SEE ALSO HEADINGTSO 1NDIVIDUAL rABLES
The list given below is intended to help candidates with formulae which
may be found hard to memorize. Inclusion in the list does not mean that
a proof may not be required.
1
1. FINITE DIFFERENCES
Newton’s formula: ux+,,,, = u~+ri~l)Au,+n(z)A2u,+. . .
Newton’s divided-difference formula :
U, = u,+(x-cI) Au.+(X-(I)(X-b) A%,+. . .
b br
Lagrange’s interpolation formula :
-
U,
(x-a)(x-b). . .(x-k)-
U, 1 + ub 1 + ...
(a-b)(a-c). . .(a-k)’x-a (6-a)(b-c). . .(b-k)’x-b
Gauss’s forward formula:
+
U, = UO +x~~)Au~+x(~,A1 +~(ux+- 1)(3)A3u- 1 +(x+ l),,) . . .
Gauss’s backward formula :
U, = UO +x(l)A U- 1 +(x + I)(,) AZu- 1
. .
+ ( X + ~ ) ( ~ ) A ~ U ~ ~ + ( X + ~ ) ( ~ , A ~ ~ - ~ + .
Summation by parts: 1I ( U, Au,) = [ux 0,]1’ -Ea (U,+ Au,)
1 1
(l+A)’ E &= _du, -- A uX-- A%, +-A-% , ..*
dx 2 3

Negative There are two formulations, one typically for Subject I
Binomial and one typically for Subject 5.
For subject 1 :
Parameters: k, a positive integer; 0 < p < 1 with q = 1- p
(iI : )
‘, + +
PF. P(X = X) = pkqx- x = k, k 1, k 2, . . .
L
k kq
E(X) = -, Var(X) = -
P P2
For subject 5:
Parameters: k > 0; 0 < p < 1 with q = 1 -p
(“
PF. P(X = x) = +;-l)Pv. x = 0, I, 2,
”,,)”
PGF. G(s) = -
(I
k9 kq
E(X) = -, Var(X) = -
P P2
The two formulations are of course connected, each
differing from the other by a shift in location. In
particular, if X, is as in the subject 1 formulation and X,
as in the subject 5 formulation with an integer value fork,
+
then X, = X, k.
Geometric Negative Binomial with k = 1
2.4. CONTINUOUS DISTRIBUTIONS
Normal Parameters: - cc < p < 00, 0 > 0
( a
+
MGF. M(t) = exp pf - a2t2
E(X)= p, Varw) = n2
1
Gamma . Parameters: c! > 0, A > 0
. *
-
AaXa I
PDF.f(x) = -e pAx,x > O
r(a)
( ;)-’
MGF.M(t)= I-- ,t<E.
Exponential Parameter: 1 > 0
PDF. f(x) = Ae-A“, x > 0
E(X) = 1 /i.V,a r(X) = I/;?
n 1
Chi-square 12 is Gamma with a = -2 and I = -2.
where n is a positive integer
Beta Parameters: a > 0, j > 0
MGF.
a
E(X) = -c! + p’ V ar(X) = (a+ B>’(a + P + 1)
Lognormal Parameters: - CO < p < CO, CJ > 0
-i(%)2}
exp{
PDF.f(x) = ,x>o
XCJ&
MGF.
-
3
E(X)= e.xp {p + -0' , Var(X) = exp(2p + a'}. [exp{a2)- I]
L
Pareto Parameters: 2 > 0, ,I > 0
+
PDF. f@) A ai"(l x)-~-',x > 0
MGF.
-
E(X)= L/(a- I), Var(X) = a,12/{(a- I)'(a-2)}
Generalised Parameters: tl > 0, E . > 0, k > 0
Pareto +
T(cr k)l"xk
PDF.f(x) = + - I x > o
r(a)r(k)(i x ) ~ + ~ '
MGF
-
+
E[X]= ik/(a- I), Var[X] = i2k(k a- I)/{(a- I)'(cr-2)}
Weibull Parameters: c > 0, y > 0
PDF.f(x) = cyx'-lexp{ -cxy}, x > 0
MGF.
-
Burr Parameters: a > 0, ,I > 0, y > 0
+
PDF. f(~=) a yi"xS-'(EL xY)-. I, x > 0
MGF.
-
2.5. COMPOUND- D ISTRIBUTIONS
Compound N Poisson (A), {X,}zI i i d.
Poisson
N
Y = X,(=OifN=O)
I= 1
MGF. of Y. M,,(t) = exp{l(M,(r)- 1))
where M,(t) is the MGF of X,
L
2.6. PARAMETRIC INFERENCE, NORMAL MODEL
The random sample (x,, x2, . ?2 , k,) has mean R and variance
cx2
- (Cx)2/n
32 =
n- 1
(a) Fo-r a single sample of size n under the normal model
X N(P, a*)
-
(n- 1)S2
(ii) x,-
1
U2
(b) For two in-d ependent sam-pl es of sizes rn and n under the normal
models X N(px ,,): a Y N(p ,,,a’,> respectively
(ii) under the additional assumption :a = a’,