Question

Use a random number generator in Python (e.g. numpy.random.uniform()) and use it to approximate the integral I = int_0^2 (x^(4)+ 2x^(3) − 2x)dx , (1) using an increasing number of n points. Repeat this calculation m = 100 times at each value of n, starting with n = 100 points and increasing linearly up to n = 5000 in steps of 100. Each of these evaluations will produce an estimate ˆIn,i for each of the i evaluations for a sample size of n. At each sample size, calculate the mean squared error (MSE) of the approximation: MSE_(n) =(1/m)sum_(i=1)^m (I_(n,i)-I)^(2), (2) where I is the true value of the integral. Plot how the MSE changes as a function of n (this plot does not need to be turned in, but is useful for your understanding). At what value of n does the MSE drop below 1?

Name: use a random number generator in python eg numpyrandomuniform and use it to approximate the integral i int02 x4 2x3 2xdx 1 using an increasing number of n points repeat this calculation m 1 64493
Uploaded: 2022-11-09T17:01:50-08:00
Duration: 2 min 26 s
Channel: Kirsty Gledhill
Description: use a random number generator in python eg numpyrandomuniform and use it to approximate the integral i int02 x4 2x3 2xdx 1 using an increasing number of n points repeat this calculation m 1 64493

          Use a random number generator in Python (e.g. numpy.random.uniform()) and use it to approximate the integral
I = int_0^2 (x^(4)+ 2x^(3) − 2x)dx , (1) using an increasing number of n points. Repeat this calculation m = 100 times at each value of n, starting
with n = 100 points and increasing linearly up to n = 5000 in steps of 100. Each of these evaluations
will produce an estimate ˆIn,i for each of the i evaluations for a sample size of n. At each sample size, calculate the mean squared error (MSE) of the approximation:
MSE_(n) =(1/m)sum_(i=1)^m (I_(n,i)-I)^(2), (2)
where I is the true value of the integral. Plot how the MSE changes as a function of n (this plot does
not need to be turned in, but is useful for your understanding). At what value of n does the MSE drop
below 1?

Added by Victoria B.

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