# Random Numbers and Probability Distributions

I/

Explain why a decision maker might feel uncomfortable with the expected value approach, and decide to use a non-probabilistic approach even when probabilities are available.

II/

1. ________________ is a technique for selecting numbers randomly from a probability distribution.

2. Random numbers of a mathematical process instead of a physical process are _______________ numbers.

3. The drying rate in an industrial process is dependent on many factors and varies according to the following distribution.

Minutes Relative Frequency

3 0.22

4 0.36

5 0.28

6 0.10

7 0.04

Using these random numbers, simulate the drying time for 5 processes: 0.13; 0.09; 0.19; 0.81; and 0.12.

4. Demand Frequency Random Numbers: 62 13 25 40

0 0.15

1 0.30

2 0.25

3 0.15

4 0.15

If a simulation begins with the first random number, what would the first simulation value be?

5. Demand Frequency Random Numbers: 62 13 25 40

5 0.15

6 0.30

7 0.25

8 0.15

9 0.15

Determine the random number ranges for the above data set (Start with 00).

6. If the probability of an event is 0.15, what random number range specifies this properly?

7. How many 2-digit random numbers are in the range 00 - 30?

8. The drying rate in an industrial process is dependent on many factors and varies according to the following distribution.

Minutes Relative Frequency

3 0.22

4 0.36

5 0.28

6 0.10

7 0.04

Compute the mean drying time.

9. A normal distribution has a mean of 500 and a standard deviation of 50. A manager wants to simulate 2 values from this distribution, and has drawn these random numbers: -0.6 and 1.4. What are the 2 values respectively?

10. The number of cars arriving at Joe Kelly's oil change and tune-up place during the last 200 hours of operation is observed to be the following:

Number of cars arriving Frequency

3 or less 0

4 10

10 30

6 70

7 50

8 40

9 or more 0

Determine the probability distribution of car arrivals.

11. The number of cars arriving at Joe Kelly's oil change and tune-up place during the last 200 hours of operation is observed to be the following:

Number of cars arriving Frequency

3 or less 0

4 10

11 30

6 70

7 50

8 40

9 or more 0

Based on the above frequencies, two digit random numbers are used and the following random number ranges have been developed.

Number of cars arriving Random Number ranges

4 00-04

5 05-19

6 20-54

7 55-79

Using the following sequence of random numbers, simulate 6 hours of car arrivals at Joe Kelly's oil change and tune-up facility. Random numbers: 92, 44, 15,77,21,38.

12. George Nanchoff owns a gas station. The cars arrive at the gas station according to the following inter-arrival time distribution. The time to service a car is given by the following service time distribution. Using the following random number sequence: 92, 44, 15, 97, 21, 80, 38, 64, 74, 08, estimate the average customer waiting time , average teller idle time and the average time a car spends in the system.

Interarrival times (in min.) P(X) Random Numbers Service time (in minutes) P (X) Random Numbers

4 .35 00-34 2 .30 00-29

7 .25 35-59 4 .40 30-69

10 .30 60-89 6 .20 70-89

20 .10 90-99 8 .10 90-99

13. Pseudorandom numbers exhibit ______________ in order to be considered truly random.

a. a limited number of possible outcomes

b. a uniform distribution

c. a detectable pattern

d. a detectable run of certain numbers

14. A seed value is a(n)

a. steady state solution of a simulation experiment

b. number used to start a stream of random numbers

c. first run of a simulation model

d. analytic solution of a simulation experiment

15. _____________ are the values that express the state of the system being modeled at the beginning of the Monte Carlo simulation.

a. Outputs

b. Random events

c. Initial conditions

d. Random numbers

16. Random numbers generated by a ______________ process instead of a ___________ process are pseudorandom numbers.

a. physical / physical

b. physical / mathematical

c. mathematical / physical

d. mathematical / mathematical

17. A researcher wants to simulate sunny and rainy days in her town for a 3-week period. What is the minimum number of digits the student must obtain from a random number table for each observation if it rained on two-fifths of the days over the past several years at this time of the year? Assume that days can be classified historically as either sunny or rainy.

a. 1

b. 2

c. 3

d. none of the above

18. If the probability of an event is 0.18, what random number range specifies this properly?

a. 0.10 - 0.20

b. 0.20 - 0.30

c. 0.30 - 0.40

d. 0.40 - 0.50

19. How many 2-digit random numbers are in the range 00 - 10?

a. 9

b. 10

c. 11

d. 12

20. Validation of the simulation model deals with the

a. computation of the random numbers

b. determination of the random numbers

c. determination of the solution

d. sensitivity of the solution

#### Solution Preview

Quantitative Method I/

Explain why a decision maker might feel uncomfortable with the

expected value approach, and decide to use a non-probabilistic

approach even when probabilities are available.

extra-personal factors:-

* organisational constraints: geo-politics / office politics

* possible interactions with colleagues

* trying to achieve multiple goals

personal biases:-

* humans are not entirely rational and not all decisions can

be explained in terms of probability. even if one does,

one cannot be assured that the counterparty does likewise.

* lack discipline, not understand method, not conditioned to

think probability, does not believe in the method, thinks

he/she know better, too mechanical and demeaning to making

decisions this way as a human, habit / tradition

* does not understand the problem itself

* people have different _utilities_ / preferences for

different outcomes

[according to mathematician Bernoulli people do not

generally base their decisions on expected values of

outcomes (1738; St.Petersberg Paradox); also: John von

Neumann and Oskar Morgenstern (1944)]

* decision suggested by the model may be psychologically

difficult to make

* moral / legal issues

weakness of probability estimation methods itself:-

* inadequate info

* non repeatable / rare event

* too short history record a run to estimates

* can be (own or someone else's) subjective estimates

* probabilities are available but unreliable

weakness of the method itself:-

* events not separable

* events not analysed thoroughly and correctly

* most prob estimates analyse only one point in time and

one location (required scenario may be to analyse for a

period and region)

* probabilities known, but cost of the consequences may not

be known / difficult to estimate

* models do not tell us which assumptions to choose

* models do not tell us how much to invest in improving the

information input

II/

1. ________________ is a technique for selecting numbers

randomly from a probability distribution.

Ans: Sampling

2. Random numbers of a mathematical process instead of a

physical process are _______________ numbers.

Ans: pseudorandom

3. The drying rate in an industrial process is dependent on

many factors and varies according to the following

distribution.

Minutes Relative Frequency

3 0.22

4 0.36

5 0.28

6 0.10

7 0.04

Using these random numbers, simulate the drying time for 5

processes: 0.13; 0.09; 0.19; 0.81; and 0.12.

First tabulate the cumulative rel. freq.

Minutes r.f. cumulative r.f. (running total)

...

#### Solution Summary

Random Numbers and Probability Distributions are investigated. The solution is detailed and well presented.