Submission guidelines: You can type up your answers if you wish, or you can take photos of your handwritten answers and upload those.
Problem One.Consider the following situation. Engineers working for the government of Mexico have recently discovered an offshore area potentially rich in oil and gas. To raise revenue, the Mexican government decides to auction off drilling rights to private companies. The auction is structured as a first price sealed bid auction. First price means the highest bidder will win the auction. the bidders submit bids simultaneously and secretly no bidder knows what its rivals have bid until the auction is over and the winner is declared. Three in the auction and prepare to submit their bids: Exxon, Shell, and Pemex.
a) You are hired as a consultant for Exxon to help them prepare their bid. Due to the , the bids of Shell and Pemex can be thought of as . They are denoted and respectively. Using the notation of a joint cdf, write down a simple formula for the probability Exxon will win the auction if it submits a bid of 80 million dollars.
b) Suppose your strategic analysis of Shell and Pemex leads you to believe that their bids will be independent in a probabilistic sense. Furthermore you believe that and . Given this information, what is the probability Exxon will win the auction if it submits a bid of 70 million dollars?
Problem Two. Suppose and are two independent random variables. Prove that,
Problem Three. You are a consultant hired to advise a bank. The bank has made a loan and wishes for you to perform an analysis on the creditworthiness of the borrower. In particular the bank wishes to know if the borrower will default on their loan in the next hundred days.
a) Common to credit modeling, you decide to use anexponential probability distributionto describe the time to default random variable . In this type of credit analysis it is assumed that every borrower will eventually default if given enough time, and this default occurs at a time from the present. A bigger means the borrower will default later on, while a smaller means the borrower will default sooner. The exponential probability distribution gives a cdf for of,
Where the term is aparameterthat describes how creditworthy the borrower is. A higher corresponds to a riskier borrower and a lower corresponds to a safer borrower. According to your analysis, you estimate that is equal to 0.002. What then is the probability that the borrower will default in the next hundred days, i.e. that ?
b) What is the probability the borrower will default in the next two hundred days?
c) Suppose the economy goes into a sudden and due to the Covid-19 pandemic. This causes you to revise your estimate of to 0.003. How does this change your answer to part a) and b) above?
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