Q1. The so-called 2-parameter Weibull distribution is defined as follows:

F(x) = 1-e^-(x/b)a.

(i) Using the inverse transform method, derive the formula for generating random numbers for the two-parameter Weibull distribution, where a=3 and b=0.9. 

(ii) Using the formula generate a sample from u = 0.72 where u is the uniformly distributed random number.

(Please note that in the exponential term above, a is the power of x/b)

Q2. Using the following random numbers between 0 and 1, generate random numbers for the uniform distribution between 5 and 10.

0.1, 0.6, 0.9, 0.7, 0.5.

Compute the sample mean and standard deviation from the samples generated.

Q3. Generate random numbers between 0 and 1 using the random number generator:

I_j+1 = aI_j mod m.

Use a=3, m=20 and I₀=15.

(i) What is the period of this generator? Would you recommend its use in a commercial package?

(ii) What distribution does this sequence of random numbers generate?

Q4. (Monte Carlo Simulation) I generate the following sequence of random numbers between 0 and 10. 

5.5, 8.9, 6.2, 4.5, 6.5, and 7.8.

Peter uses the above to generate random values for X, which is uniformly distributed between 1 and 4, while Jane uses the same sequence to generate random values for Y, which is uniformly distributed between 25 and 36.  Compute the sample mean of (X+Y) from the values generated by Peter and Jane. 

Q5. In a single-server queue, successive inter-arrival times are: 2.1, 3.1, 1.4, 3.7, 5.0, 3.8, 2.3, and 2.9.  The service times for these queues are 5.1, 4.2, 1.1, 1.2, 1.7, 3.5, 2.1, and 3.2. Simulate the system until the exit of the 8^th customer. 

(i) Determine the sample mean of the waiting time in the queue and the mean number in the queue based on this simulation.  

(ii) Are these estimates reliable estimates of the parameters they seek to estimate?  Explain.

Q6. Match terms from Groups A and B

Group A: normal, histogram, uniform, Erlang, middle point, exponential, most likely value

Group B: waterfall, mode, rectangle, median, Bell curve, mountain, pdf/pmf

Q7. The following approximation is to be used to generate random numbers, z, from the standard normal distribution:  z = (u^0.135 - (1-u)^0.135)/0.1975

Where u is a random number from the uniform distribution, unif(0, 1). For u = 0.4, generate a value from the normal distribution x whose mean (µ) is 10 and standard deviation (σ) is 1. Note that x = µ + zσ.

Q8. The following data comes from a normal distribution:

5.6, 7.3, 5.2, 6.3, 7.2, 6.9, 7.5, and 7.6.

Compute the sample mean and standard deviation, as well as the confidence interval on the mean.

Q9. An analyst notes that the histogram of a continuous random variable, which takes values between 1 and 6, has a quadratic shape. He develops the following model for the pdf:

f(x) = (1/60)x²

Is the model correct? Why or why not? Use mathematical arguments to provide your reasons.

Q10. The following data is collected for annual incomes (in thousands) in a small population (an office). As an analyst, you are expected to provide the range as well a value that represents the middle point of the income. Please do so.

51.2, 48.9, 95.9, 125.6, 45.8, 52.3, 58.2, 61.4, 49.4, 56.1, 48.8, 44.3.