This applet demonstrates an elementary application of importance-sampling Monte Carlo integration to evaluate a simple integral.
By clicking on the appropriate button, you can evaluate the integral via either the simple Monte Carlo or importance-sampling Monte Carlo integration. In either case, the number of quadrature points Nused to estimate the integral can be varied by changing the value in the box (the largest value accepted is 1000000).
You may select from three integrands for the calculation (each is defined such that the integral evaluates to exactly 1). The "Display Function/Quadrature" button toggles between showing quadrature information and showing plots of the integrand. In the integrand plot you see the integrand itself, f(x), the weighting function w(x) used for the importance-sampling calculation, and the ratio f(x)/w(x). To the extent that this ratio is not strongly dependent on x, the importance sampling approach becomes more effective. The sin and exp functions are selected to make this ratio nearly constant with x, while it varies some when x^2 is the integrand.
The simple Monte Carlo calculation is performed by selecting N points with uniform probability on (0,1); the importance-sampling calculation samples points with a linearly-increasingly probability w(x) over the same interval. This causes more points to be taken near x = 1, which is also where the integrand has its largest value.
The result of the calculation is displayed in the quadrature view, along with the error in the result (given that the exact value of the integral is 1). Also, in one figure you will see the x-values used to perform the quadrature; note how there are more of them near x = 1 in the importance-sampling scheme. This distribution of samples can be seen more clearly by examining the histogram of quadrature values. Superimposed on the histogram is the distribution w(x) governing the sampling of the quadrature points.
Repeated pressing of the buttons generates new samples, with different x-values taken to give new estimates of the integral.