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Math Help - Monte Carlo Integration

  1. #1
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    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.

    ANSWER:
    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)?

    ANSWER:
    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.

    ANSWER:
    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.

    Thanks
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  2. #2
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    For the standard deviation, use the fact that the total number of points which fall beneath the curve has a Binomial distribution, where p = ...
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