Consider the one-dimensional integrals: a. Evaluate the integrals by Monte Carlo techniques, applying comparatively uniform and exponential sampling (using function randExp). Consider numbers of...

Consider the one-dimensional integrals:

a. Evaluate the integrals by Monte Carlo techniques, applying comparatively uniform and exponential sampling (using function randExp). Consider numbers of sampling points ranging from n = 250 to 100,000, and plot the integral estimates and the associated standard deviations.

b. Explain the significant variance reduction when using exponential sampling for the first integral and the lack of effect in the case of the second integral.

c. Use the routine LinFit ( from modfunc.py) to perform linear regression and extract the scaling laws of the standard deviations for the two probability distributions from their log–log plots.