Monte Carlo Simulator

Random Variables and Probability Distributions

A random variable is a quantity whose value is uncertain and can vary according to a probability distribution. For example, tomorrow's stock price, next month's customer demand, or the duration of a project task are all random variables—you don't know their exact values in advance, but you can describe the range and likelihood of different outcomes.

Probability distributions describe how likely different values are for a random variable. Common distributions used in Monte Carlo simulation include:

• Normal (Gaussian) distribution: Symmetric, bell-shaped curve centered around a mean with spread controlled by standard deviation. Used for many natural phenomena (heights, measurement errors, returns) and central limit theorem applications.
• Uniform distribution: Every value within a range [min, max] is equally likely. Used when you have little information beyond upper and lower bounds (e.g., "completion time is between 5 and 10 days").
• Triangular distribution: Specified by min, mode (most likely value), and max. Simple and intuitive for expert estimates where you know the range and the peak likelihood.
• Lognormal distribution: Skewed right, values are always positive (cannot be negative). Used for prices, incomes, project costs, and other quantities that grow multiplicatively.
• Exponential distribution: Models time until an event occurs (e.g., time between arrivals, equipment lifespan). Memoryless property makes it useful for queues and reliability.

Choosing the right distribution is critical. The distribution should match the real-world behavior of the uncertain variable: if demand is typically around 100 with symmetric variability, use normal; if it's bounded between 80 and 120 with no clear peak, use uniform; if it's skewed and can't be negative, consider lognormal. This tool lets you select from multiple distributions and set parameters to match your textbook problem or scenario.

What Is Monte Carlo Simulation?

Monte Carlo simulation is a technique for approximating the distribution of an outcome when that outcome depends on uncertain inputs. The process is straightforward:

1. Define your model: Specify how uncertain inputs combine to produce an outcome (e.g., profit = price × quantity − cost).
2. Assign distributions: For each uncertain input, choose a probability distribution and parameters.
3. Generate random samples: Use random number generation to draw values from each distribution. For example, if price ~ Normal(100, 10), draw a random price like 103.2.
4. Calculate outcome: Plug the sampled inputs into your model to compute the result (e.g., profit = 103.2 × 95 − 7000 = 2804).
5. Repeat many times: Do steps 3–4 thousands or millions of times, recording each outcome.
6. Analyze results: Compute summary statistics (mean, standard deviation, percentiles), visualize the distribution (histogram), and estimate probabilities (e.g., probability profit < 0).

The power of Monte Carlo is flexibility: you don't need a closed-form formula for the distribution of the outcome. The simulation approximates it numerically by generating a large sample. As the number of trials increases, the approximation improves due to the law of large numbers.

Monte Carlo trades exact analytical solutions for approximate but practical answers. For simple models, you can sometimes derive formulas for mean and variance analytically. But for complex models—nonlinear relationships, multiple interacting uncertainties, custom logic—simulation is often the only feasible approach.

Key Outcome Metrics and Statistics

After running a Monte Carlo simulation, you'll see several key statistics that summarize the distribution of outcomes:

• Mean (Expected Value): The average outcome across all simulation trials. This is an estimate of the true expected value of your model. For example, if mean profit = $5,000, that's the long-run average result if you repeated this scenario many times.
• Median: The middle value when outcomes are sorted. Half of simulations are above, half below. Median is robust to extreme outliers and gives a sense of the "typical" outcome.
• Standard Deviation / Variance: Measures how spread out the outcomes are. High standard deviation means high variability and risk; low standard deviation means outcomes are clustered near the mean.
• Percentiles: Values below which a certain percentage of simulations fall. For example, the 5th percentile is the value below which 5% of outcomes lie (a worst-case estimate), and the 95th percentile is the value below which 95% of outcomes lie (a best-case threshold). Percentiles are critical for risk assessment: "There's a 10% chance we'll earn less than $2,000 (10th percentile)."
• Probability of Events: Estimate the likelihood of specific outcomes. For example, P(profit < 0) is the fraction of trials where profit was negative, giving you the probability of a loss. Similarly, P(outcome > target) tells you how often you exceed a goal.
• Minimum and Maximum: The lowest and highest outcomes observed in the simulation. These give a sense of the extreme scenarios but are sensitive to the number of trials (more trials may uncover more extreme values).

These metrics connect to core statistics concepts: the law of large numbers says that as the number of trials increases, the simulated mean converges to the true expected value. The central limit theorem ensures that sampling error decreases with more trials. Monte Carlo provides an empirical distribution—based on simulated data—that approximates the theoretical distribution of your outcome.

Mode 1: Single-Distribution Simulation

Use this mode to explore a single random variable and understand its distribution through simulation.

1. Add a random variable: Click "+ Add" if needed, or use the default variable.
2. Choose a distribution type: Select from Normal, Uniform, Triangular, Lognormal, or Exponential.
3. Enter parameters:
   • Normal: Mean (center) and Standard Deviation (spread)
   • Uniform: Min and Max (range boundaries)
   • Triangular: Min, Mode (most likely), and Max
   • Lognormal: μ (mean of log) and σ (std dev of log)
   • Exponential: λ (rate parameter)
4. Set output expression: For a single-variable exploration, use just the variable name (e.g., "demand").
5. Set number of trials: Start with 10,000 for quick results; increase to 100,000 or more for smoother distributions.
6. Click "Run Simulation".
7. Review results: Summary statistics (mean, median, std dev), percentiles (P5, P10, P25, P75, P90, P95), histogram showing distribution shape, convergence plot showing how the mean estimate stabilizes.

Use this mode for: Verifying textbook distribution properties, building intuition about shapes (normal vs uniform vs skewed), understanding how sample size affects estimation accuracy.

Mode 2: Custom Formula / Multi-Variable Model

Use this mode to model complex scenarios with multiple uncertain inputs.

1. Define each uncertain input:
   • Name your variables clearly (e.g., "price", "demand", "cost").
   • For each variable, select a distribution and set parameters.
   • Add as many variables as needed using "+ Add".
2. Enter your output expression: Type a formula combining your variables. Examples:
   • Profit model: price * quantity - cost
   • Project duration: task1 + task2 + task3
   • Net present value: revenue / (1 + discountRate) - investment
   • Use JavaScript Math functions: Math.max(demand - inventory, 0) * unitCost
3. Set number of trials: 10,000 is typical; use more for smoother results or complex models.
4. Click "Run Simulation".
5. Review results:
   • Mean outcome and standard deviation (risk measure).
   • Percentiles for best-case/worst-case scenarios (e.g., 5th percentile = downside risk).
   • Histogram showing full distribution of possible outcomes.
   • Sensitivity analysis (if supported): correlations between input variables and outcome, showing which inputs drive the most variability.

Use this mode for: Business case analysis (profit/revenue/cost models), project risk assessment (schedule/budget), inventory/demand planning, portfolio return modeling (homework-style), any scenario with multiple uncertain factors.

Mode 3: Scenario Comparison (if UI supports it)

Compare multiple parameter sets side-by-side to understand trade-offs.

1. Set up Scenario 1 (e.g., "Conservative"): Define distributions for all inputs with cautious parameters (lower means, higher variability if risk-averse).
2. Run simulation and save or note results.
3. Set up Scenario 2 (e.g., "Optimistic"): Adjust parameters (higher means, lower variability if confident).
4. Run simulation again.
5. Compare: Side-by-side summary statistics, overlapping histograms, differences in downside risk (5th percentile) and upside potential (95th percentile).

Use this mode for: Sensitivity to assumptions, strategic planning (conservative vs aggressive strategies), communicating range of possibilities to stakeholders.

General Tips for Using the Simulator

• Start with more trials: 10,000 is good for quick exploration; 100,000+ for smoother, more stable estimates. More trials reduce sampling error.
• Use the seed for reproducibility: The "Seed" input ensures the same random sequence if you re-run. Change it to get a different set of random draws.
• Check convergence plots: If the mean estimate still "wiggles" a lot at the end, you may need more trials.
• Match distributions to reality: Don't just default to normal—choose the distribution that best represents the actual variability (uniform for bounded unknowns, lognormal for skewed positive quantities, triangular for expert estimates with a peak).
• Interpret percentiles carefully: The 5th percentile is not "the worst possible outcome" but rather "the value below which 5% of simulations fall." It's a risk threshold, not a guarantee.
• Remember scope: This is a learning and exploration tool. Use results to inform decisions, not as promises or guarantees. Always validate assumptions and consult domain experts for high-stakes choices.

Core Sampling and Aggregation Algorithm

The Monte Carlo simulation process follows a simple iterative algorithm:

After N simulations, compute summary statistics from the collected outcomes:

• Mean (Expected Value): mean ≈ (1/N) Σ y_i
• Variance: var ≈ (1/(N−1)) Σ (y_i − mean)²
• Standard Deviation: stdDev = √var
• Percentiles: Sort {y_1, y_2, ..., y_N} and pick values at ranks corresponding to desired percentiles (e.g., 5th percentile at rank 0.05×N).
• Probability of event: P(y < threshold) ≈ (number of trials where y_i < threshold) / N

The approximation improves as N increases due to the law of large numbers: the sample mean converges to the true expected value, and the sample distribution approaches the true distribution of the outcome.

Law of Large Numbers (Conceptual)

The law of large numbers is the theoretical foundation of Monte Carlo simulation. It states that as the number of independent trials N increases, the sample average (mean of simulated outcomes) converges to the true expected value of the random variable.

Practical implications:

• With small N (e.g., 100 trials), the simulated mean may differ noticeably from the true mean due to sampling error.
• With large N (e.g., 100,000 trials), the simulated mean is very close to the true mean.
• Convergence plots show this visually: as N increases, the running average stabilizes.
• There is always some simulation error, but you can make it arbitrarily small by increasing N.

Similarly, the central limit theorem ensures that the sampling distribution of the mean becomes approximately normal as N increases, allowing us to construct confidence intervals for our estimates.

Worked Example 1: Simple Normal Distribution Simulation

Worked Example 2: Profit Model with Uncertainty

1. Homework in Probability or Statistics Course

Scenario: A textbook problem asks you to compute the probability that a sum of two independent uniform random variables exceeds a threshold. The analytical solution is tricky, but you can verify your answer using Monte Carlo.

How this tool helps: Define two uniform variables, set the output expression as their sum, run 10,000+ simulations, and compute the empirical probability. Compare to your analytical result. If they match, you're confident. If not, review your work. Monte Carlo provides a sanity check and builds intuition for distributions of sums and products.

2. Finance / Portfolio Risk Example (Educational)

Scenario: A finance course problem models a simple portfolio with two assets. Each asset has uncertain annual returns modeled as normal distributions. You want to estimate the distribution of total portfolio value after one year and the probability of losing money.

How this tool helps: Define variables for each asset's return (e.g., asset1 ~ Normal(0.08, 0.15), asset2 ~ Normal(0.06, 0.10)), set the output expression as a weighted portfolio value (e.g., 50% each), run simulations, and review the histogram and percentiles. Estimate P(return < 0) to quantify downside risk. This is a conceptual exercise for learning risk analysis, not actual investment advice.

3. Project Management and Schedule Risk

Scenario: A project consists of three tasks in sequence. Task durations are uncertain: Task1 ~ Triangular(5, 7, 10 days), Task2 ~ Triangular(3, 5, 8 days), Task3 ~ Uniform(4, 6 days). What is the probability the project finishes within 18 days?

How this tool helps: Define three variables for task durations with their distributions, set output expression = task1 + task2 + task3, run 50,000 simulations. Review the distribution of total duration and compute P(duration ≤ 18). If the probability is low (say, 30%), you know there's significant schedule risk and should plan buffers or contingencies. This is a simplified PERT-style analysis using Monte Carlo.

4. Inventory / Demand Planning

Scenario: A retailer wants to decide how much inventory to stock for a seasonal product. Demand is uncertain: Normal(mean = 200, std dev = 40). Each unit costs $10 to stock, sells for $30. Unsold units are discarded (or heavily discounted, effectively $0 value). What inventory level maximizes expected profit?

How this tool helps: Use Monte Carlo to simulate profit for different inventory levels. For each level S, define profit = min(demand, S) × 30 − S × 10. Run simulations for S = 150, 180, 200, 220, 250, etc. Compare mean profits to find the optimal S. This is a newsvendor-style problem where simulation reveals the profit distribution for each strategy, helping you balance understocking vs overstocking risk.

5. Engineering Reliability or Quality Margin (Educational)

Scenario: A classroom engineering problem models a component's strength as Lognormal(mean_log = 5, std_log = 0.3) and the applied load as Normal(mean = 100, std = 20). The component fails if load > strength. Estimate the probability of failure.

How this tool helps: Define variables for strength (lognormal) and load (normal). Output expression: load - strength (or use indicator load > strength ? 1 : 0). Run simulations and compute P(failure) as the fraction of trials where load exceeded strength. This builds intuition for safety margins and reliability analysis, foundational for engineering risk assessment.

6. Scenario Testing for Exam Prep

Scenario: You're preparing for a statistics or operations research exam where you'll need to explain Monte Carlo methods, interpret simulation results, and discuss convergence. You want to run multiple examples quickly to build deep understanding.

How this tool helps: Set up different distribution combinations (normal + uniform, triangular + exponential), change parameters, vary the number of trials (1,000 vs 10,000 vs 100,000), observe how percentiles stabilize, and practice interpreting histograms and convergence plots. Rapid iteration with this tool sharpens your intuition and prepares you to explain concepts clearly under exam pressure.

7. Business Case Analysis: Comparing Strategies

Scenario: A business case assignment asks you to compare two pricing strategies. Strategy A: high price ($80) but lower demand (Uniform 50–80 units). Strategy B: low price ($60) but higher demand (Uniform 90–130 units). Fixed costs = $2,000. Which strategy is better in terms of expected profit and risk?

How this tool helps: Run two simulations. Simulation 1: price = 80, quantity ~ Uniform(50, 80), profit = 80 × quantity − 2000. Simulation 2: price = 60, quantity ~ Uniform(90, 130), profit = 60 × quantity − 2000. Compare mean profits, standard deviations, and downside risk (5th percentile). Strategy with higher mean profit might have more variability (higher risk). This analysis informs strategic recommendations and demonstrates risk-return trade-offs.

8. Understanding Law of Large Numbers and Sampling Error

Scenario: A conceptual learning exercise: you want to see the law of large numbers in action. How does increasing the number of simulations improve accuracy?

How this tool helps: Set up a simple model (e.g., mean of a normal distribution). Run 100 trials, note the simulated mean. Run 1,000 trials, note the mean. Run 10,000, then 100,000. Watch the convergence plot: the running average gets closer to the true mean and stabilizes. This visual demonstration reinforces the theoretical concept and shows why "more trials = better estimates," a key principle in Monte Carlo and statistics.

1. Using Unrealistic or Inconsistent Distributions

Mistake: Modeling a quantity that cannot be negative (like demand, time, or price) with a normal distribution that allows negative values.

Why it matters: If the distribution has significant probability mass below zero, simulation results will include impossible outcomes (negative demand, negative duration), distorting summary statistics.

How to avoid: Choose distributions carefully. For strictly positive quantities, use lognormal, exponential, or truncate normal to exclude negatives. For bounded quantities, use uniform or triangular. Match the distribution to the real-world behavior and constraints of the variable.

2. Mixing Units or Scales

Mistake: Modeling revenue in dollars but costs in thousands of dollars, or durations in different time units (hours vs days), without converting to a common unit.

Why it matters: The output expression will mix incompatible units, giving nonsensical results. For example, profit = revenue (in dollars) − cost (in thousands) would understate costs by a factor of 1,000.

How to avoid: Before entering distributions, define units clearly for all variables (e.g., all money in dollars, all time in days). Convert inputs as needed. Double-check the output expression to ensure units are consistent (dollars − dollars = dollars, hours + hours = hours).

3. Running Too Few Simulations

Mistake: Running only 100 or 1,000 trials and treating the results as highly accurate or making decisions based on noisy estimates.

Why it matters: With few trials, sampling error is large. The simulated mean may differ noticeably from the true mean, and percentiles are unstable. Re-running the simulation gives different results each time.

How to avoid: Use at least 10,000 trials for preliminary exploration, and 50,000–100,000+ for smoother, more stable results. Check the convergence plot: if the mean estimate still fluctuates a lot near the end, increase the number of trials. Remember: more trials = smaller sampling error.

4. Ignoring Model Assumptions and Limitations

Mistake: Forgetting that simulation results depend entirely on the chosen distributions, parameters, and model structure. Treating outputs as "objective truth" rather than "estimates based on assumptions."

Why it matters: If your distributions don't match reality (e.g., you assume normal when actual data is skewed), results are misleading. There's no hidden "magic"—garbage in, garbage out.

How to avoid: Always state your assumptions explicitly. Validate distribution choices with historical data or expert judgment. Run sensitivity analyses: change parameters slightly to see how results shift. Present results with caveats: "Assuming demand is Normal(100, 15), we estimate..." not "Demand will definitely be..."

5. Misreading or Misinterpreting Percentiles

Mistake: Confusing "95th percentile = $10,000" with "there's a 95% chance the result equals exactly $10,000," or thinking the 5th percentile is "the absolute worst case that could ever happen."

Why it matters: Percentiles are thresholds, not point probabilities. The 95th percentile means "95% of simulations are at or below this value," not "95% of the time you'll get this exact number." The 5th percentile is a downside risk measure, but even lower outcomes are possible (just rare).

How to avoid: Interpret percentiles correctly: "There's a 5% chance the outcome will be below $2,000 (5th percentile)" or "There's a 90% chance the outcome will fall between the 5th and 95th percentiles." Use ranges, not single points, to communicate uncertainty.

6. Treating Approximate Results as Guarantees

Mistake: Using Monte Carlo outputs as promises or guarantees of future performance. For example: "The simulation says mean profit is $5,000, so we'll definitely earn $5,000."

Why it matters: Simulation provides estimates based on modeled uncertainty, not certainties. Real outcomes depend on many factors not captured in a simple model.

How to avoid: Frame results as estimates and ranges: "We expect mean profit around $5,000, with 90% confidence the outcome will fall between $3,000 and $7,000." Emphasize that simulation supports decision-making but doesn't eliminate risk or uncertainty. This is especially critical in finance, where results should never be treated as investment guarantees or trading advice.

7. Forgetting About Dependencies Between Variables

Mistake: Modeling price and demand as independent when they're actually correlated (e.g., higher price reduces demand). Ignoring correlation leads to unrealistic scenarios in the simulation.

Why it matters: If variables are truly correlated in reality but modeled as independent, the simulated outcome distribution will be too wide or too narrow, giving misleading risk estimates.

How to avoid: Think carefully about relationships between inputs. If correlation exists, either use distributions that respect it (if the tool supports correlated sampling) or adjust your model logic (e.g., make demand a function of price). For basic homework and conceptual learning, independence is often assumed for simplicity, but recognize the limitation.

8. Ignoring Convergence and Not Checking Stability

Mistake: Running a simulation once and accepting the first result without checking if the estimate is stable. Re-running gives a noticeably different answer, indicating insufficient trials.

Why it matters: With too few trials or high-variance models, results can vary significantly between runs, reducing confidence in conclusions.

How to avoid: Use convergence plots to visually assess stability. Run the simulation multiple times (with different seeds) and compare results. If they're similar, you have confidence. If they differ a lot, increase the number of trials. Reporting a confidence interval for the mean (based on standard error) is also good practice in formal analysis.

9. Over-Interpreting Tail Outcomes (Min/Max)

Mistake: Focusing heavily on the absolute minimum or maximum observed in the simulation and treating them as "true worst/best case scenarios."

Why it matters: Min and max are highly sensitive to the number of trials. With more trials, you'll likely see even more extreme values. They don't represent the range of all possible outcomes, just the range observed in this sample.

How to avoid: Use percentiles (5th, 95th) instead of min/max for risk assessment. Percentiles are more stable and representative of "likely worst/best cases" rather than "absolute extremes." If you need to communicate worst-case planning, use low percentiles with appropriate caveats.

10. Not Documenting Assumptions and Parameters

Mistake: Running simulations without writing down which distributions and parameters you used, making it impossible to reproduce results or explain them to others.

Why it matters: For homework, reports, or presentations, stakeholders need to know what assumptions drove the results. Without documentation, results are not credible or reproducible.

How to avoid: Always document: which variables, which distributions, parameters (mean, std dev, min, max, etc.), number of trials, seed (for reproducibility), and the output expression. Include this in your write-up or presentation. Good documentation makes your analysis transparent, credible, and replicable.

1. Sensitivity Analysis: Vary One Parameter at a Time

Change one distribution parameter (e.g., increase the standard deviation of demand by 20%) while holding others constant, then re-run the simulation. Compare results to see how sensitive the outcome is to that input. This identifies which uncertainties matter most and where better estimation or risk mitigation should be focused.

2. Focus on the Tails: Downside and Upside Scenarios

Don't just report the mean—emphasize the 5th percentile (downside risk), 95th percentile (upside potential), and the probability of specific events (e.g., P(loss), P(exceeding target)). Tail analysis is critical for risk management. Communicate results as: "On average we expect $X, but there's a 10% chance we'll earn less than $Y (10th percentile)." This frames uncertainty clearly for stakeholders.

3. Scenario Design: Build Multiple Parameter Sets

Create "Conservative," "Base Case," and "Optimistic" scenarios with different distribution parameters. Run simulations for each and compare the full distributions, not just means. This shows how outcomes shift under different assumptions and helps stakeholders understand the range of possibilities. Scenario comparison is a powerful tool for strategic planning and communicating uncertainty.

4. Communicating Results: Use Ranges and Probabilities

Express results as ranges and probabilities instead of single-point predictions. Instead of "Profit will be $5,000," say "Expected profit is $5,000, with 80% confidence between $3,500 and $6,500." Use histograms and percentile charts in presentations to visually communicate uncertainty. This transparency builds trust and helps decision-makers understand risk.

5. Combining Analytical and Simulation Approaches

Use Monte Carlo as a double-check for analytical formulas learned in class. For simple models (e.g., sum of normals is normal with combined variance), verify simulation results match theory. For complex models without closed-form solutions, rely on simulation but understand the underlying statistical principles. This hybrid approach strengthens both your analytical and computational skills.

6. Understand and Leverage the Law of Large Numbers

Run the same simulation with increasing numbers of trials (1,000; 10,000; 100,000) and watch how the mean stabilizes. This visual demonstration of the law of large numbers is a powerful learning tool. In practice, use enough trials that adding more doesn't change the mean noticeably—this ensures your estimate is converged and reliable.

7. Exploring Correlation and Dependency Effects

If your tool supports correlated sampling, experiment with positive and negative correlations between variables. For example, model price and demand with negative correlation (high price reduces demand). Observe how correlation affects the width of the outcome distribution. Understanding dependency is crucial for realistic models in finance, operations, and risk analysis.

8. Using Convergence Diagnostics for Quality Control

Check convergence plots and run diagnostics: Does the running mean stabilize? Are percentiles consistent across multiple runs? If not, increase trials. For very high-stakes analyses (not typical for this educational tool), you might compute confidence intervals for the mean using standard error formulas. This quality control ensures simulation results are trustworthy.

9. Practice with Diverse Models to Build Intuition

Run simulations for varied problem types: profit models, project schedules, inventory decisions, reliability failures, portfolio returns (conceptual). The more scenarios you simulate, the faster you'll recognize patterns, choose appropriate distributions, and interpret results. Practice is the best way to master Monte Carlo and build the intuition needed for exams and real-world applications.

10. Use Monte Carlo as a Communication and Teaching Tool

Monte Carlo is not just for finding answers—it's a framework for explaining uncertainty, risk, and probabilistic thinking to others. Use simulation results to facilitate discussions: "Here's what happens if demand is higher/lower" (show histogram), "There's a 20% chance we'll miss the deadline" (percentile). Visualization and probabilistic language make complex trade-offs transparent and help align teams around data-driven decisions. This communication value is as important as the numerical results themselves.