Monte Carlo Integration
We have the function f(x) = x^2 and the integral of f(x) from 1 to 2.
Imagine that a rectangle is formed with a base of x=1 to x=2 and height y=0 to y=4.
There are 100 points randomly selected in this rectangle. 61 of them fall below the curve of f(x)=x^2.
A) Using the graph (I decribed above), calculate the monte carlo estimate for the integral of f(x) = x^2 from 1 to 2.
Fraction of randomly selected points = Area under Curve divided by rectangle R
So we have (61/100)*[(2-1)*(4-0)] = 2.44
Next question. B) What is the random sampling error of our estimate in part A)?
I figure we just do 2.44 which is the estimated value and subtract the actual area which can be found doing the actual integral. So 2.44 - 2.33333333333 = 0.1067
C) Calculate the standard error for the Monte Carlo Estimate of the integral of f(X)=x^2 from 1 to 2 when using n=100 points chosen at random from within the rectangle with base x=1 to x=2 and height y=0 to y=4.
I know that standard error is Standard deviation over sqrt of n. But how do i get the SD?
Please help complete C) and check A) and B) for correctness.
For the standard deviation, use the fact that the total number of points which fall beneath the curve has a Binomial distribution, where p = ...