Mathematical economics:

Problem 2. (8 pts) Recall the relation between degrees Fahrenheit and degrees Celsius 160 degrees Celsius . degrees Fahrenheit- 9 Let X and Y be the daily high temperature in degrees Fahrenheit for the summer in Los Angeles and San Diego. Let T and S be the same temperatures in degrees Celsius. Suppose that Cov(X, Y) = 4 and p(X, Y) = 0.8. Compute Cov(T, S) and p(T, S) (p(T, S) = correlation)Problem 1. (20 pts: 4,4,4,8) We asked 6 students how many times they rebooted their computers last week. There were 4 Mac users and 2 PC users. The PC users rebooted 2 and 3 times. The Mac users rebooted 1, 2, 2 and 8 times. Let C be a Bernoulli random variable representing the type of computer of a randomly chosen student (Mac = 0, PC = 1). Let R be the number of times a randomly chosen student rebooted (so R takes values 1,2,3,8). (a) Create a joint probability table for C and R. Be sure to include the marginal probability mass functions. (b) Compute E(C) and E(R). (c) Determine the covariance of C and R and explain its significance for how C and R are related. (A one sentence explanation is all that's called for.) Are R and C independent? (d) Independently choose a random Mac user and a random PC user. Let M be the number of reboots for the Mac user and W the number of reboots for the PC user. (i) Create a table of the joint probability distribution of M and W, including the marginal probability mass functions. (ii) Calculate P(W > M). (iii) What is the correlation between W and M?Problem 3. (16 pts: 8,8) I have a bag with 3 coins in it. One of them is a fair coin, but the others are biased trick coins. When flipped, the three coins come up heads with probability 0.5, 0.6, 0.1 respectively. Suppose that I pick one of these three coins uniformly at random and flip it three times. (a) What is P(HTT)? (That is, it comes up heads on the first flip and tails on the second and third flips.) (b) Assuming that the three flips, in order, are HT, what is the probability that the coin that I picked was the fair coin? (Remember, there is no need to simplify fractions.)