Formulas

Question 1

MGF

Answer 1

E(e^tx) = Mx(t)

Question 2

Franchise deductible in terms of regular deductible

Answer 2

E((X-d)+) +dS(d)

Question 3

TVaR

Answer 3

E(X | X > VaRp(X)) =

VaRp(X) + (E(X) - E(X min VaRp(X))/(1-p)

Question 4

Tail Weight Measures

Answer 4

1. more positive moments -> lower tail weight
2. if lim S1(x)/S2(x) > 1 or lim f1(x)/f2(x) > 1 then numerator has higher tail weight
3. increasing h(x) -> lighter tail
4. increasing ex(d) -> heavier tail

Question 5

Consistency

Answer 5

theta hat is consistent if:

1. lim pr( | theta hat - theta| < delta ) = 1 for all delta > 0, or
2. bias -> 0 and Var (theta hat) -> 0

Question 6

Cov (Fx, Fy - Fx)

Answer 6

= -Fx(Fy-Fx)/n, x< y

Question 7

variance of exact exposure

Answer 7

var(qj) = (1-qj)^2 * dj/ej^2

Question 8

var of actuarial exposure

Answer 8

qj(1-qj)/(ej/n)

Question 9

percentile matching with incomplete data: censored/truncated

Answer 9

censored -> select percentiles within the range of uncensored observations

truncated -> match percentiles of the conditional distribution

Question 10

MLE of grouped data btw d and cj and left-truncated from below at d:

Answer 10

(F(cj)-F(d))/S(d)

Question 11

MLE = MOM

Answer 11

Poisson
Binomial
NB (r known)
Gamma (a known)
Normal mean/SD

Question 12

Hypothesis tests - fitted distribution with deductible

Answer 12

F*(x) = 1- S(x)/S(d)

Question 13

5 points about K-S

Answer 13

1. only for individual data
2. lower critical value if u < infinity
3. If params are fitted, critical value should be lowered
4. Larger sample size has lower critical value
5. Uniform weight on all parts of distribution

Question 14

5 points about Chi-Sq

Answer 14

1. May be used for individual or grouped data
2. No adjustments on critical value if u < infinity
3. If parameters are fitted, critical value is automatically adjusted
4. Critical value is independent of sample size
5. Higher weight on intervals with low fitted probability

Question 15

Loss functions

Answer 15

Type of loss/bayesian estimate
squared error/mean
absolute/median
zero-one/mode

Question 16

lambda k

Answer 16

sk = -ln(1-uk)/lambda k

poisson = lambda
binomial = -mln(1-q) + k ln(1-q)
NB = r ln(1+B) + k ln (1+B)

sum from 0 - n until sum > 1, result is n.

‘time between’ type questions.

Question 17

Polar method

Answer 17

v = 2u - 1
s = v1^2 + v2^2 <= 1 then T = sqrt(-2ln(S)/S)