A market trader has the option for one day of selling either ice-cream [all], hot food {:12} or umbrellas [d3 } at an outdoor festival. He believes that the weather is equally likely to be ne {9.}, overcast {5'1} or wet {6'3} and estimates his prots under each possible scenario to be: ii} iii} {iii} {in Determine the minim solution to this problem. [2] The trader's partner is very optimistic and believes that the criterion to adopt in deciding which product to sell should be to maximise the maxim prot. Determine the decision the trader's partner would make based on these predicted prots. [1] Detemiine the Bayes criterion solution to this problem. [2] The trader's partner agrees that it is equally likely to be either ne or wet but believes that there is more than an evens chance of it being overcast. By sketching, a graph of the Bayes risk For each of the three possible decisions against the probability of it being overcast {p}, or otherwise, determine the mised Bayes criterion solution. [4] [Total 9] A statistician has to decide on the basis of a single observation from a Bin(1,8) distribution whether the value of O is %, 1/ or % . The loss suffered in the event of a wrong decision is 200 times the absolute value of the error made. (i) Draw up a table showing the 9 possible values of the loss function. [2] Identify the nine possible decision functions, and construct a table showing the values of the corresponding risk functions. [8] (iii) Show that three of the decision functions are inadmissible, and that according to the minimax criterion two of the remaining decision functions are equally good. [4] (iv) If the three possible parameter values are equally likely, which decision function(s) is preferred under the Bayes criterion? [2] [Total 16]