1. A random variable is distributed with the probability density f(x) = kx for x from the interval [0,1] and f(x) = 0 for all other x. Find: (a) the value k, (b) the cumulative distribution function for x, (c) the expected value of x, and (d) the probability that x does not exceed 0.5.
2. A computer is connected to two printers. If at least one of them works, we can still print. The time-to-repair of each printer is exponentially distributed with the mean value of 1 year. What is the probability that after 1 year, we will still be able to print?
3. For a printer-producing company, the probability that a printer needs repairs is 10%. In a sample of 100 printers, what is the expected value of number of faulty printers, and what is the standard deviation? What is the probability that no more than 15 printers are faulty (just give a formula, it is not necessary to get a numerical answer).
4. Two independent tasks are coming to the server. Their arrival times are uniformly distributed between noon and 5 pm. Compute the expected time of the task which arrives first, and the expected time of the task which arrives second.
5. Derive a formula and explain how to generate a random variable with the density f(x) = 0.5x on the interval [0,2] if you have a standard random number generator which produces values uniformly distributed on the interval [0,1]. Use the inverse transform method.
6. Suppose that we have a distribution in which we have 1 with probability θ and −1 with probability 1 − θ. Use method of moments and maximum likelihood to estimate the parameter θ from the following sample: −1, 1, −1, 1, −1, 1, −1, −1.
7. In a shipment of 100 printers, 10 turns out to be faulty. Construct a 90% confidence interval for the proportion of faulty printers.