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["        LLN Uniform Random Variable"]
[]
[]
["Left endpoint",{"$f":"3*PI()","v":9.4247779607693793},null,"Enter data in yellow cells and function formula in B10. Outputs appear in B12 and B13."]
["Right endpoint",{"$f":"4*PI()","v":12.566370614359172}]
["Sample size",1000]
["No. of iterations (even)",200]
[null,null,null,"           Illustrates the law of averages for iid random variables of the form f(X),"]
["x (random point)",null,null,"        where X is a uniform random variable on the interval [LPoint, RPoint] and f"]
["f(x)",{"$f":"SIN(B9)","v":0},null,"       is a user defined or Excel function entered in cell B10 with argument B9"]
[null,null,null,"       (for example, = B9^3 or = SIN(B9)). Sample averages are gotten by executing"]
["Sample average",-0.63585233476970715,null,"       the number of runs in B6, each run consisting of a randomly generated number"]
["Expectation (using Simpson)",-0.6366197065536322,null,"       in [LPoint, RPoint] which appears in cell B9. The runs are averaged and the "]
[null,null,null,"       result appears in cell B12. The expectation of the random variable f(X),"]
[null,null,null,"       computed by Simpson's rule, is given in B13 for comparison. Cell B7 gives"]
[null,null,null,"       the number of iterations used in Simpson's rule. The larger the number, the "]
[null,null,null,"       more accurate the integral approximation. Also, the larger the sample size,"]
[null,null,null,"       the closer the sample average is to the expectation."]

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