Marks for each question are indicative. More marks than the total stated
might be awarded for exceptional creativity, presentational clarity or computing
1. Use a simulation method to calculate the integrals∫ 1
− exp
and ∫ 1
∫ 1
exp(− exp(xy)) dxdy
as accurately as you can.
2. An actuary is using the following model for the proportion of settled claims
after time t. The proportion is X where
P(X ≤ x) = xtet(1−x).
Time is measured in small units. This is not a stochastic processes model;
the actuary is just modelling the marginal distribution of the proportion of
settled claims at a fixed time. Generate samples from the distribution of X
for various values of t. Note that X is a proportion so it takes values between
0 and 1. Please read the following question before deciding which values of t
to use.
3. Continuing form the previous question, a mathematician discovered that the
random variable Z = t(log(X))2 is approximately exponentially distributed
with mean 2 for large values of t. She did that by calulating the limit of
P(Z > z) as t → ∞. The actuary was sceptical about the approximation
overall and in particular she claimed that the approximation was perform-
ing especially badly for large values of Z (corresponding to small values of X).
Investigate all arguments using a sample generated in the previous question
and suggest an improvement by choosing a different approximating distri-
bution. No answer that does not reproduce the mathematician’s calculations
can be regarded as complete.
4. Consider a pack of 10 cards marked 1,2,3,3,4,4,5,6,7,8 (3 and 4 are repeated).
There are two players that chose 3 cards each (a total of 6 cards without
replacement are drawn). The player with the lowest total wins and the loser
pays the winner an amount equal to the smallest card the winner is holding.
If the totals are equal there is no winner. So if player A has drawn 2,3 and
5 and player B 1,4 and 7, A is the winner and B pays her an amount of
2. Let Y be the amount that A wins or loses by after each turn. Find the
distribution of Y .
5. Continuing from the previous question, the rules are modified a bit. Player
A has the right to choose one of the three cards in advance and the other 5
cards are drawn from the remaining pack. Choosing 1 is tempting because
it maximises the chances of winning but then every time she wins she will
only win by 1. She is thinking of choosing card 2 as the chances of winning
are still good and the winning amount will be 2 more often than 1. Also the
possible losing payments will be a bit lower. Can she do even better? The
game is neither fair nor symmetric any more.