Chapter 4_Monte Carlo in Applications (Enumeration)

Stanislaw Ulam,

John von Neumann

Pioneers of Monte Carlo Simulation (2)

There is uncertainty or randomness;

Many possible outcomes exist;

The problem is difficult to solve exactly;

We want to estimate probability or risk

Reasons to Use Monte Carlo Methods (4)

Randomness;

Probability;

Uncertain outcomes

Elements That Make Monte Carlo a Stochastic Simulation (3)

Random values are generated;

The values are used as inputs;

The model is evaluated;

The process is repeated many times

Random Sampling Process (4)

Define the problem/model;

Specify input variables;

Generate random inputs/samples;

Run the model or experiment (simulation);

Collect/aggregate results;

Analyze the output;

Interpretation and decision making

Monte Carlo Algorithm Steps (7)

Finance;

Healthcare;

Engineering;

Weather and climate studies;

Risk analysis;

Business;

Transportation;

Scientific research

Applications of Monte Carlo Methods (8)

Possible profit;

Possible loss;

Investment risk /

Outcomes Estimated in Finance Application (3)

Number of patients;

Waiting times;

Required staff;

Possible resource shortages

Factors Estimated in Healthcare Application (4)

Rainfall;

Wind speed;

Flood level;

Population exposure

Conditions Simulated in Disaster Risk Application (4)

Finishing early;

Finishing on time;

Finishing late

Project Completion Probabilities Estimated (3)

Uses formulas;

May produce an exact answer;

Best for simpler problems;

Usually deterministic

Traditional Approach Characteristics (4)

Uses repeated random trials;

Produces an estimate;

Useful for uncertain or complex problems;

Usually stochastic

Monte Carlo Approach Characteristics (4)

Handle uncertainty;

Examine many possible outcomes;

Support risk analysis;

Can be applied to complex problems;

Provide estimates when exact solutions are difficult

Advantages of Monte Carlo Methods (5)

Provide estimates, not guaranteed exact answers;

Depend on the quality of assumptions;

May require many trials;

Can produce misleading results if incorrect inputs are used

Limitations of Monte Carlo Methods (4)

Random Number Generation;

Law of Large Numbers;

Central Limit Theorem

Core Principles of the Monte Carlo Method (3)

Weak LLN;

Strong LLN

Law of Large Numbers types (2)