1. In the U.S., racial profiling describes the practice of law enforcement agencies to search, stop and sometimes arrest people of a particular ethnic group more than their relative number in the population. Suppose that a population in a certain city is composed of 30% belonging to ethnic group 1 and 70% to ethnic group 2. Members of the ethnic groups are visibly different. Court records reveal that crime rate in group 1 is 25% and in group 2 it is 10%. A police officer stops a person at random. Let G1 be the event that the person belongs to group 1, G2 the event that the person belongs to group 2 and C the event that the person is a criminal.
(a) What is the probability that the person is a criminal?
(b) What is the probability that the person is from group 1 if he is a criminal?
(c) What is the probability that the person is from group 2 if she is a criminal
2. students were asked 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. (2 points)
(b) Compute E(C) and E(R). (2 points)
(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? (2 points)
(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. (2 points) (ii) Calculate P(W > M). (2 points) (iii) What is the correlation between W and M? (2 points
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