Simulation Modeling

Monte Carlo Simulation

Monte Carlo Simulation is a type of probabilistic model. probabilistic: Processes with randomness and uncertainty involved. Same input may have different output. deterministic: Processes with no randomness and uncertainty involved. Same input always has same output.

Simulating Deterministic Behavior: Area Under a Curve

In the graph, $y=f(x)$, where $0\le f(x)\le M$ and $a\le x\le b$ . Now we select a random point $P(x,y)$ in the rectangle. Create two counters, one to count the total points and the other to count the points in the region below the curve. The formula to calculate the approximate value for the area below the curve is:

$$\text{area under curve}\approx \frac{\text{number of points counted below curve}}{\text{total number of random points}}\cdot \text{area of rectangle}$$

This method needs a large number of trails. In general, to cut the expected error in half, four times more experiments are needed.

Volume Under a Surface

The method we use to approximate the volume is similar to the method we use to find the area under a curve. Here is the formula:

$$\text{volume of object to be measured}\approx \frac{\text{number of points in the object to be measured}}{\text{total number of points}}\cdot \text{total volume}$$