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Risk Modeling & Probability Distributions Guide

Understanding the mathematical foundations behind Monte Carlo simulations. Learn how different probability distributions model real-world uncertainty and risk.

📊 Mathematical Foundations of MonteSim.ai

Common Probability Distributions

Fixed Distribution
Constant value with no uncertainty

Best Used For:

Known constants, regulatory requirements, fixed costs

Example:

Monthly rent ($2,500), tax rate (15%), contract duration (12 months)

Formula:

f(x) = value (constant)
Uniform Distribution
Equal probability across a range

Best Used For:

When all values within a range are equally likely

Example:

Random delivery times between 2.5-4.8 days, temperature variations (68.2-71.7°F)

Formula:

f(x) = 1/(b-a) for a ≤ x ≤ b
Discrete Uniform Distribution
Equal probability for integer values in a range

Best Used For:

Counting scenarios, whole number outcomes

Example:

Number of team members (3-8), dice rolls, survey ratings (1-5)

Formula:

P(X=k) = 1/(b-a+1) for k ∈ {a,a+1,...,b}
Triangular Distribution
Most likely value with symmetric or asymmetric spread

Best Used For:

When you have minimum, most likely, and maximum estimates

Example:

Project duration (min: 30 days, likely: 45 days, max: 60 days)

Formula:

Peak at mode, linear decrease to min/max
Normal Distribution
Bell curve with mean and standard deviation

Best Used For:

Natural phenomena, measurement errors, large sample averages

Example:

Human heights, test scores, manufacturing tolerances

Formula:

f(x) = (1/σ√2π)e^(-½((x-μ)/σ)²)

Advanced Distributions

Bernoulli Distribution
Binary outcomes (success/failure)

Use Case:

Yes/no scenarios, pass/fail events

Example:

Product launch success (70% chance), equipment failure (5% chance)

Formula:

P(X=1) = p, P(X=0) = 1-p
Log-Normal Distribution
Positively skewed, cannot be negative

Use Case:

Stock prices, income distributions, project costs

Example:

Software development costs, stock returns, real estate prices

Formula:

ln(X) ~ Normal(μ, σ²)
Beta Distribution
Bounded between 0 and 1, flexible shape

Use Case:

Percentages, probabilities, completion rates

Example:

Market share (0-100%), project completion percentage

Formula:

f(x) = x^(α-1)(1-x)^(β-1)/B(α,β)
Exponential Distribution
Time between events, decay processes

Use Case:

Waiting times, equipment lifespans, customer arrivals

Example:

Time between customer calls, component failure times

Formula:

f(x) = λe^(-λx) for x ≥ 0

Risk Modeling Concepts

Correlation Modeling
How variables influence each other

MonteSim.ai can model positive, negative, or no correlation between variables to create realistic scenarios.

Sensitivity Analysis
Which inputs matter most

Automatically identifies which variables have the greatest impact on your outcomes.

Scenario Planning
Multiple what-if analyses

Run different scenarios to understand best-case, worst-case, and most likely outcomes.

Risk Quantification
Measure uncertainty precisely

Get specific probabilities for different outcome ranges, not just gut feelings.

How MonteSim.ai Applies These Concepts

🤖 AI-Powered Distribution Selection

Our AI analyzes your scenario description and automatically selects the most appropriate probability distributions for each variable. No need to be a statistics expert!

  • • Recognizes keywords and context clues
  • • Suggests realistic parameter ranges
  • • Handles complex multi-variable scenarios

📊 Intelligent Correlation Modeling

MonteSim.ai understands that real-world variables don't exist in isolation. It models realistic relationships between your inputs.

  • • Detects logical correlations (cost vs. quality)
  • • Prevents unrealistic combinations
  • • Creates more accurate simulations

🎯 From Description to Distribution in Seconds

1

Describe Your Scenario

Write in plain English about your business situation

2

AI Selects Distributions

Our AI chooses the right mathematical models

3

Get Results & Insights

Receive professional analysis and recommendations

Ready to Apply These Concepts?

Let MonteSim.ai handle the complex mathematics while you focus on making better decisions. Start with your free credits today.

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