1. In a dark, dystopian future caused by a certain politician who thought the big red button on his desk was to order more cheeseburgers, the world is suffering from the consequences of nuclear disaster. Like many others, you have started growing your own vegetables in your garden as the food supply is unreliable. However the amount you can harvest and safely eat is also affected by the fallout and is hard to predict with great certainty due to the unstable circumstances. You have constructed a model, in which the amount of edible harvest (in kgs) per year is modelled by a Gamma(2, ) distribution for some unknown parameter > 0 that represents the pollution level in your garden and vegetables. Your initial belief was that this parameter follows an Exp(1) distribution. During the past three years, you have been able to harvest resp. 5, 7 and 3 kgs of edible vegetables. (a) Determine the posterior distribution after these observations. Hint: you should find one of the well known distributions. [5 marks] (b) There are Greengrocers in this world that you decide to sell your latest harvest to. These Greengrocers have a Geiger counter that allow them to measure the pollution level in your vegetables. The higher the pollution level they measure, the less they pay you. However you have learned from your old nan that rubbing some marmite on your vegetables tricks the Geiger counters into reporting an incorrect pollution level, which can be lower but also higher than the actual level. To be precise, if the true pollution level is > 0 and you rub on a 0 teaspoons of marmite, then the price you get paid per kg of vegetables is given by (a + 1)e (a+1) . Compute the optimal number of teaspoons of marmite you should rub on a kg of vegetables (note that this can be any positive real number, not necessarily an integer). Hint: any integrals you encounter can be computed very quickly using the attached list of distributions! [7 marks]

3. Consider the capital process (Ut)t0 in the Cramer-Lundberg model, where time t is measured in years. Consider the following two probabilities: (i) the probability that ruin happens during the first year, (ii) the probability that U1 < 0. Without doing any computations, which of these two is larger? Briefly motivate your answer. [2 marks] 4. Trading on financial markets is nowadays commonly done by computer systems that make largely autonomous decisions based on models and algorithms rather than the classic image of a crowd of women and men on an exchange floor shouting their buy and sell orders erratically into their phones! Suppose that a particular system is aimed at making a specific type of deals. The investment (in ) that it needs to do to enter such a deal depends on variable market conditions and can be modelled by an Exp(1) distribution, while the gain (in ) can be modelled by an Exp(1.1) distribution. It is assumed that the investment and the gain are independent of each other, and also that the deals happen independently of each other. The resulting profit (positive or negative) of such a deal is the gain minus the investment. Suppose that the system runs for a year, but that it trades so quickly and so often that for convenience we may assume that it does infinitely many deals. We also assume that it is allowed to continue trading no matter how much money it has lost. If the system has a starting capital of 10, what is the probability that at any point during the year the system has a capital of more than 20? Hint: bit of a challenge! ;). However this question has a very natural (and interesting!) link to the Cramer-Lundberg material we have discussed. Once you realise what the link is, the working to compute the requested probability is pretty quick. [4 marks]