## ats606/

# ats_mc_integrate_example.pro

ATS606, Stats, Integration, Monte Carlo

includes main-level programIn ATS606, we discuss how to integrate a function that is not analytically integrable. This function provides an example how this done, using the normal distribution.

The normal distribution, of course, is given by $$ f(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2} $$

To find the probability a random draw from the distribution falls between $a,b$ we integrate: $$ Pr = \int_a^b f(x) dx = \int_a^b \frac{1}{\sqrt{2\pi}} e^{-x^2/2} dx $$

This is not, in general, integrable. But, the integral can be estimated with Monte Carlo methods. In general, an integral is estimated with: $$ I \approx (b-a)/n * \sum_i f(x_i) $$

where $a,b$ are the lower and upper limits of the integration, and $f(x)$ is the function you wish to integrate. The $x_i$ are random values of the integrand which are derived from the uniform distribution.

## Examples

To estimate the integral of the normal distribution betwen -1 and 1, which we know should yield $0.683$, try this:

```
n = 1e2 ;number of random samples to use
xMin = -1 & xMax = 1
int = ATS_MC_INTEGRATE_EXAMPLE(n, xMin=xMin, xMax=xMax)
print, int
```

## Author information

- Author
Phillip M. Bitzer, University of Alabama in Huntsville, pm.bitzer "AT" uah.edu

- History
Modification History:

`First written: Apr 9, 2014`

## Routines

## top ats_mc_integrate_example

`result = ats_mc_integrate_example( [num] [, xMin=float] [, xMax=float])`

This estimates the integral of the normal distribution betwen the given limits. On error, a NaN is returned.

### Return value

The value of the integral of the normal distribution between the given limits.

### Parameters

- num in optional type=integer default=100
The number of samples used to estimate the integral.

### Keywords

- xMin in optional type=float default=-1.
The lower limit of the integration

- xMax in optional type=float default=1.
The upper limit of the integration

## File attributes

Modification date: | Wed Apr 16 12:06:44 2014 |

Lines: | 27 |

Docformat: | rst rst |