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1 Definition Exercice 1. 1. Write a function Fhat(t,x) which returns the empirical distribution at t given an n-sample x. Here the function takes one real number and one vector. Vectorizing the fuction: Fhat now takes two vectors, which is useful for next questions. 2. Consider x an n-sample from a normal distribution N(0,1) with curve representing the distribution function F. n = 100. Plot F̂n together with the  2 Convergence of the empirical distribution function Exercice 2. Consider x an n-sample from an exponential distribution ε(1) and t = 2. 1. Observe the convergence of F̂n(t) to F(t) by comparing both values for the following values of n = 10, 30, 50, 100, 200 and 500. There is a simple convergence, in fact, even if the values aren't always decreasing, a global attenuation of the gap is noticeable (between the cdf approximation and the cdf) when n increases. Those results can be explained by the fact that the sample always changes. 2. Central Limit theorem in action. (a) Using the function qqnorm, check that √𝑛{F̂n(t) ̶ F(t)} has (approximatively) a normal distribution N(0,F(t){1 ̶ F(t)}) for the previous values of n. Quick maths: Var(N(0,F(t){1 ̶ F(t)})) = √F(t){1 ̶ F(t)}*Var(N(0,1)) but qqnorm compares Sample Quartiles to N(0,1)'s Theoretical Quartiles so we have to compute: n = 10, 30, 50, 100, 200, 500 and t = 2. √n{F̂n(t) ̶ F(t)} √F(t){1 ̶ F(t)} with  Sample Quantiles ≈ Theoretical Quantiles therefore distribution N(0,F(t){1 ̶ F(t)}). √𝑛{F̂n(t) ̶ F(t)} has approximatively a normal