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.J 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 circurustances. You have constructed a model. in which the amount of edible harvest (in kgs} per year is modelled by a Garmna[2,] distribution for some unknown parameter 6' :r- {1 that represents the pollution level in your garden and vegetables. TYour initial belief was that this parameter follows an Exp} distribution. During the past three years1 you have been able to harth resp. 51 T and 3 kgs of edible vegetables. [a] Determine the posterior distribution after these observations. Hint: you should find one of the Twell knownT 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 level1 which can be lower but also higher than the actual level. To be precise, if the true pollution level is H :3 [l and you rub on a E 1'] teaspoons of marmite1 then the price you get paid per kg of vegetables is given by (a + 1]e["+\"9. Compute the optirual 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}