## Statistics Spring 2014 Test 2

Name:
1. A random variable is distributed with the probability
density f(x) = kx^{2} 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 2
years. What is the probability that after 4 years, we will
still be able to print?

3. For a printer-producing company, the probability that a
printer needs repairs is 1%. In a sample of 500 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 20 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 8 am and noon.
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) = (3/8) x^{2} 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. In a computer, 1 is represented by a 10V signal, and 0 by a
5V signal. Suppose that we have a distribution in which we have
10 with probability θ and 5 with probability 1 −
θ. Use method of moments and maximum likelihood to
estimate the parameter θ from the following sample: 10,
5, 5, 10, 5, 5, 10, 10.

7. In a shipment of 100 printers, 10 turns out to be faulty.
Construct a 90% confidence interval for the proportion of
faulty printers.