Let X be a random variable with a distribution function: si means when a) derivate F (by hand) in order to calculate the density of X b) Calculate E(X) and Var(X) (by hand) and provide details of your calculation c) For all y _ (0, 1), calculate F_1 (y) d) Use your answer in c) to write a code on MATLAB that simulates 100 000 random variables for which the distribution function is given by F e) Estimate E(X) and Var(X) on MATLAB and compare to answers found in b) 2. Let X be a random

View complete question Let X be a random variable with a distribution function: si means when a) derivate F (by hand) in order to calculate the density of X b) Calculate E(X) and Var(X) (by hand) and provide details of your calculation c) For all y _ (0, 1), calculate F_1 (y) d) Use your answer in c) to write a code on MATLAB that simulates 100 000 random variables for which the distribution function is given by F e) Estimate E(X) and Var(X) on MATLAB and compare to answers found in b) 2. Let X be a random variable with a density : si means when ; Sinon means otherwise a) Find the density g of the uniform law [_1, 1] and a constant c* > 0 optimal (smallest value possible) that satisfies f(x) _ c*g(x) for all x _ [_1, 1]. In other words find : Provide details and demonstrate that the critical point found is in fact a point that maximizes b) Using the reject algorithm and the constant c* found in a), write a MATLAB code that simulates a vector of 100 000 random variables that have F as density c) Estimate E(X) and Var(X) using MATLAB 3. a) Explain (by hand) how to use Monte Carlo simulation to estimate the following integral : b) Taking n = 100 000, write a MATLAB code that estimates the integral and give a confidence interval with a confidence level of 95% c) Find the real value of the integral using the integral function on MATLAB and compare it with the answer found in b

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