Monte Carlo simulation repeatedly samples uncertain inputs to produce a distribution of possible outcomes. Instead of one fixed forecast, it shows how often different results occur. It is useful when interacting variables or nonlinear relationships make an exact solution impractical, and when a decision depends on risk as well as the average outcome. section 9.0 section 9.1
A practical workflow is:
- Define uncertain inputs and their distributions using historical data, domain knowledge, or explicit assumptions. Check that the chosen distributions fit the quantities being modeled. section 9.2
- Generate outcomes. For discrete values, cumulative probabilities can map random numbers to outcomes. For continuous stock returns, estimated parameters can drive draws from a normal distribution. section 9.2 section 9.5.1
- Repeat the trial, count or plot the results, and compare decisions against the resulting variability—not just the mean. Use enough trials to avoid leaning on an anomalous small sample. section 9.3.5 section 9.3.6
Staffing for absenteeism
Suppose a call center’s daily absentee count follows a Poisson distribution with an estimated rate of two. Ten simulated days produce a mean of 2.4 absences, while the calculated expected value is about 2. That difference is not evidence that the rate has changed: ten draws can be unrepresentative. A manager can weigh extra payroll against the service risk of having too few staff; 500 automated trials give a clearer view of the modeled distribution. section 9.3.1 section 9.3.6 section 9.4
>>> np.random.seed(1) #1
>>> num_trials = 500 #2
>>> normalized_probabilities = \
>>> probabilities / probabilities.sum() #3
>>> simulated_values = np.random.choice(k_values, #4
>>> size = num_trials, #5
>>> p = normalized_probabilities) #6
>>> unique, counts = \
>>> np.unique(simulated_values, #7
>>> return_counts = True) #8
>>> simulated_counts = \
>>> dict(zip(unique, counts)) #9
>>> simulated_percentages = \
>>> {k: (v / num_trials) * 100 for k, \
>>> v in simulated_counts.items()} #10
The Python implementation seeds the random generator, normalizes probabilities over the included absentee counts, draws 500 values with NumPy, and tabulates their frequencies. Normalization matters because higher counts were omitted from the displayed set. section 9.4
For stock prices, historical closing prices instead supply daily log returns. Their estimated mean and standard deviation parameterize simulated daily returns, producing a range of potential price paths rather than a single forecast. Such results remain conditional on the historical data and the assumed return distribution. section 9.5.1 section 1.4.2
