Write a C++ program that randomly samples from the Watt Fission Spectrum.
Say you have a Monte Carlo neutronics code. Part of this code will involve randomly sampling what energy a new neutron created from fission will have. This can be done using the Watt distribution. For this problem, we will assume the probability distribution of the energy is
P(E) = 0.4865 sinh(√2E) e−E
1. Setup.
a. First create a lambda function fx within main() which is passed E and returns the value of P(E) from the equation above.
b. Check that you have fx correct by integrating using trapezoidal from [0,8] and showing your result is 1.00 (accurate to about the hundredth place). Print the integral to the screen.
2. Find the maximum of fx ( P(E) ).
a. You can do this either by finding the roots of the derivative of fx (ie using finite_difference_derivative and bisect) and evaluating fx at the root, or by using minima. Print the (E, P(E)) to the screen.
3. Random point.
a. Randomly create a point in the range x=[0,8] and y=[0,max from part 2]. You will need to randomly select two values. Create this as a pair called point.
4. Use rejection method.
a. Check to see if the point selected in part 3 is under the P(E) curve. If it is, save the energy to a vector of doubles called sample . If it is not, reject the point and randomly create a new one and check again. Repeat until a valid point is found and the selected energy is stored.
5. Create a sample.
a. Randomly sample 100,000 energies from the Watt Fission Spectrum. They should be stored in sample . Print the size of sample to the screen.
6. Binning.
a. Create a vector of 100 energy bins, from 0 to 8 MeV, called bins . Find the number of energies in sample that are in each bin. Print to the screen the number of selections found in each of the 100 bins.
7. Histogram plot.
a. Make sure your random sampler works by plotting a histogram of sample, by plotting the value in each bin of bins. You can use Excel, Python, C++, etc. If not in your C++ code, then attach the other file with the image of the plot.
Attach all relevant code and screenshots.
