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 Cramr-Lundberg material we have discussed. Once you realise what the link is, the working to compute the requested probability is pretty quick.