SHEET 7

Using Monte Carlo Technique to Sample from Discrete Distributions

For each question: develop the corresponding code (by any computer language or

Excel) . Then solve it using paper, pencil and calculator.

1. Given the following 10 random numbers:

0.95, 0.02, 0.60, 0.44, 0.93, 0.86, 0.42, 0.19, 0.35, 0.94

Use theses random numbers to generate random observations for throwing a dice.

Plot the histogram and the associated discrete probability function.

Compare the two plots.

What should you do to enhance the resemblance of the two plots?

2. Use the same 10 random numbers (as in problem 1) to generate random samples

of “inter-arrival time” based on the following table:

Time (min) 1 2 3

Prob. 0.25 0.5 0.25

Plot the histogram and the associated discrete probability function.

Compare the two plots.

What should you do to enhance the resemblance of the two plots?

3. Given the following 10 random numbers:

0.26, 0.83, 0.01, 0.93, 0.27, 0.15, 0.60, 0.83, 0.20, 0.34

Use these random numbers to generate random samples of “service time” based on

the following table:

Time (min) 1 2 3

Prob. 0.5 0.25 0.25

Plot the histogram and the associated discrete probability function.

Compare the two plots.

What should you do to enhance the resemblance of the two plots?

4. By utilizing the “inter-arrival time” samples associated with the second question,

and the “service time” samples associated with the third question, solve the

corresponding simple one server queuing system

Find the percentage utilization of the server.

Find the average waiting time in the queue.

5. Use the same 10 random numbers (as in problem 1) to generate random

observations of the colors of a traffic light found by a randomly arriving car, when

green is 60% of the time, yellow is 10% and red is 30%.

Plot the histogram and the associated discrete probability function.

Compare the two plots.

What should you do to enhance the resemblance of the two plots?

6. Use the Monte Carlo Method to estimate the reliability of the following system

(i.e. the probability of the success of the system):

An industrial plant is protected by a system that consists of a sensor, an actuator,

and a shutdown valve arranged in series. The probabilities of failure for the units

are 10%, 5% & 2% respectively. All three units must perform satisfactory if the

system is to operate correctly.

Use the following random numbers for the sensor:

0.95, 0.02, 0.60, 0.44, 0.93, 0.86, 0.42, 0.19, 0.35, 0.94

And use the following random numbers for the actuator:

0.26, 0.83, 0.01, 0.93, 0.27, 0.15, 0.60, 0.83, 0.20, 0.34

And use the following random numbers for the valve:

0.13, 0.76, 0.07, 0.70, 0.49, 0.07, 0.27, 0.95, 0.82, 0.78

7. The weather can be considered a stochastic process because it evolves in a

probabilistic manner from one day to the next. Suppose that for a certain city the

weather forecast can be described as follows:

• The probability of rain tomorrow is 0.6 if it is raining today.

• The probability of being clear tomorrow is 0.8 if it is clear today.

Forecast the weather for the next 10 days; assuming that today is a clear day.

Hint: Use the same random numbers as in problem 1.