Question:

Two tasks, taking 30 minutes each, are scheduled to run on a server to start at any time between 4:00 am and 6:00 am, uniformly at random and independently of each-other. If one task is running and the other task starts, then the server crashes.

(a) Draw a picture of the region of joint times which would crash the server. What is the probability that the server crashes?

(b) Given that the server crashes and that one of the tasks starts at 5:30 am, what is the conditional pdf of the time at which the other task starts? Clearly specify where it is zero.

2. Let U and V be independent and identically distributed Ber(p) random variables.

Define X = min { U, V} and Y = max {U, V}.

(a) Write down the joint pmf (as a table) of X and Y . Are X and Y independent?

(b) Determine the marginal pmf of X and the conditional pmf of Y given X = 1.

3. Suppose that a database contains ten thousand items. Two quality control personnel, Xavier and Yvonne, operate independently of each-other, and uncover an error in each item with probability 0.09 and 0.1, respectively, independently of all other errors uncovered.

Denote by X the number of errors Xavier uncovers, and the number of errors Yvonne uncovers by Y .

(a) Determine the expectations and variances of X and Y.

(b) Using the central limit theorem and the table of the standard normal cdf, approximate the probability that Xavier uncovers more errors than Yvonne.

4. Repeatedly (10³ times) simulate realisations of the number of errors noticed by Xavier and Yvonne. Plot a histogram of the dierence in number of errors noticed, and use crude Monte Carlo to estimate the probability that Xavier uncovers more errors than Yvonne. Supply a typical estimate, histogram, and your code.

5. Let X₁, .........., X_N be a random sample from an exponential distribution Exp(λ).

(a) Determine the maximum likelihood estimator λ^{^} for λ.

(b) Via the central limit theorem, determine an approximate 1 - α stochastic confidence interval for 1/λ based on λ^{^}.

6. Repeatedly (10⁴ times) simulate 100 realisations from an Exp(1) distribution and record the maximum likelihood estimator 1/λ^{^} for 1/λ. For each simulation, record whether or not the true mean of 1/λ = 1 falls within the approximate 95% confidence interval constructed in Question 5. Plot a typical empirical cdf of 1/λ^{^}, a typical proportion of simulations for which the true mean lies outside the approximate confidence intervals, and supply your code.