note roboticus <p>[BrowserUk]:</p> <p>In the iterative solution, we're accumulating f(x)=(1-e^(-x/N))^h over the range of x=[0 .. NumSamples]. That's a rough form of computing the definite integral of the expression. Integrating over three variables (x, N, h) would be a pain, so I treated N and h as constants.</p> <p>So first, we multiply out our f(x) expression to remove the exponent. So using h as 2, we get:</p> <c> f(x) = 1 - 2e^(-x/N) + e^(-2x/N) integ(f(x)) = integ( 1 - 2e^(-x/N) + e^(-2x/N) ) = integ( 1 ) - 2*(integ(e^(-x/N))) + integ(e^(-2x/N)) since: integ(1) = x + C integ(e^(bx)) = (1/b)e^(bx) + C integ(f(x)) = [ x+C ] + [ (-2N)e^(-x/N) + C ] + [ (-N/2)e^(-2x/N) + C ] </c> <p> Computing a definite integral over a range is simply integ(f(x)) evaluated at the upper limit less the value evaluated at the lower limit. This causes the C terms to cancel. </p> <p>Pascal's triangle comes out because we've got (a+b)^n, and when we multiply it out, we get the binomial expansion which is where the coefficients come into play.</p> <p>One point I should mention: You don't have to use 0 as the lower bound. If you wanted the number of collisions you'd experience from sample A to sample B, just evaluate integ(f(B))-integ(f(A)). By using A=0 we compute the number of collisions for the entire run.</p> <p>...[roboticus]</p> <p><i>When your only tool is a hammer, all problems look like your thumb.</i></p> 962802 963234