Consider the following distribution:

Demand

Frequency

0

0.15

1

0.30

2

0.25

3

0.15

4

0.15

Random numbers are represented by ri, where 0

Question

Consider the following distribution:

Demand

Frequency

0

0.15

1

0.30

2

0.25

3

0.15

4

0.15

Random numbers are represented by ri, where 0 <= ri <= 1

i = 1, 2, 3, 4, 5, …

Please also consider this following sequence of random numbers:

r1 = 0.36; r2 = 0.79; r3 = 0.75; r4 = 0.54; and r5 = 0.72

Please indicate which one of the following options would represent the demand value in case the simulation begins with the first random number.

BlackTom AI · Accepted Answer

Cumulative probability mapping is used to convert a uniform random number into a discrete demand value by partitioning the interval [0,1] according to the given frequencies. 1) Relevant knowledge points and concepts - A discrete probability distribution can be transformed into cumulative intervals on [0,1]. - A random number r in [0,1] is mapped to the demand whose cumulative interval contains r. 2) Step-by-step reasoning - Start with the probabilities (frequencies) for each demand: - P(D=0) = 0.15 - P(D=1) = 0.30 - P(D=2) = 0.25 - P(D=3) = 0.15 - P(D=4) = 0.15 - Compute cumulative probabilities in order to form intervals. 3) Intermediate calculations (cumulative sums) - Cumulative after D=0: 0.15 -> Interval for 0: [0.00, 0.15) - Cumulative after D=1: 0.15 + 0.30 = 0.45 -> Interval for 1: [0.15, 0.45) - Cumulative after D=2: 0.45 + 0.25 = 0.70 -> Interval for 2: [0.45, 0.70) - Cumulative after D=3: 0.70 + 0.15 = 0.85 -> Interval for 3: [0.70, 0.85) - Cumulative after D=4: 0.85 + 0.15 = 1.00 -> Interval for 4: [0.85, 1.00] 4) Why the correct answer is correct - The first random number r1 = 0.36. Check which interval contains 0.36: 0.36 ∈ [0.15, 0.45). That interval corresponds to demand = 1. Therefore r1 maps to demand value 1. 5) Why the other options are incorrect - 0 would require 0 ≤ r < 0.15; but 0.36 is greater than 0.15. - 2 would require 0.45 ≤ r < 0.70; 0.36 is below 0.45. - 3 would require 0.70 ≤ r < 0.85; 0.36 is below 0.70. - 4 would require 0.85 ≤ r ≤ 1.00; 0.36 is well below 0.85. 6) Conclusion - The demand corresponding to r1 = 0.36 is 1, which matches the provided correct answer: 1.

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