XY = np.column_stack((X,Y))

Sample_XY = np.random.choice(XY,1,replace=True)

Sample_X = Sample_XY [0]

Sample_Y = Sample_XY [1]

Index = np.arange(0,len(X))

Sample_index = np.random.choice(Index,1,replace=True)

Sample_X = X[Sample_index]

Sample_Y = Y[Sample_index]

Sample_X = np.random.choice(X,1,replace=True)

Sample_Y = np.random.choice(Y,1,replace=True)

The core of Monte Carlos methods is repeated random sampling

Monte Cardo methods can be used to verify statistical/probablity theories.

Any numerical procedure involving random sampling can be called Monte Carlo method

Monte Cardo methods can be used to estimate the sensitivites of a calculation to random errors.

When realizing Monte Carlo in coding, the random sampling/sample generation code should be outside the for loop

Monte Carlo method is a numerical method

Monte Cardo methods can be used to estimate random output of a system with random inputs.

Monte Carlo is an analytical method.

In sample with replacement, each drawn sample will be put back to the population.

In sample without replacement, each drawn sample will be put back to the population.

In sample without replacement, we have chance to reselect some of the selected samples

In sample with replacement, we have chance to reselect some of the selected samples

In sample without replacement, we have NO chance to reselect some of the selected samples

In sample without replacement, each drawn sample will NOT be put back to the population.

In sample with replacement, each drawn sample will be NOT put back to the population.

In sample with replacement, we have NO chance to reselect some of the selected samples