questions for calculus, statistics and probability

Question 12 3 pts Suppose that X is a random variable with a mean of 6 and variance of 8; ) is a random variable with a mean of -10 and variance of 18; and Cov(X, Y) = -8. Find Corr(X, Y). O -1/3 O -1/2 O -2/3 O -3/4 O -4/5 Question 13 3 pts Given an exponential random variable X ~ Exp(X = 1/180), what is the conditional probability P(X > 100|X > 40)? O e-1/3 O e-2/9 O e-1/9 O e-1/18 01-e-2/9Consider a circle inscribed in a unit square (so that the circle has area 7/4). We toss 1000 darts randomly into the square, and it turns out that 784 of those darts land in the circle. Using the method described in class, estimate . 0 3.136 O 3.156 O 3.208 O 3.422 O 3.568 Question 19 3 pts Suppose we use Monte Carlo integration to approximate I = f2 en(2x - 4) dx. If U1, U2, . .., Un are i.i.d. Unif(0, 1 )random numbers, which of the following is a good approximation In for I? On En(60.) O _Et in(60; -4) " Lin en (60; - 4) " Li en (60; - 2)Question 20 3 pts Consider the following event list for a FIFO single-server queue, where i is the customer's sequence number, A; is the ith customer's arrival time, T, is the customer's service start time, and S, is the customer's service time. A; -JN9 8 3 10 12 4 15 19 21 22 Event list for FIFO What is the average time in system (service + waiting time) for all 5 customers? O 1.6 O 2.4 O 3.4 O 4.6 O 6.0