1 Approximations to the Binomial DistributionLetn = 10, 50, 100, 500, 1000, 5000,andp = 0.001, 0.003, 0.01, 0.03, 0.1, 0.3, 0.5, 0.7, 0.9, 0.99, 0.999.For each pair (n, p) above, and 0 ≤ k ≤ 10, use built-in R functions to tabulate P(X = k) for X ∼ Bin(n, p) andthe most appropriate approximation based on the Poisson or NormalDistributions. For each exact probability p and approximate probability p'calculate the relative error |p−p'|/pand include this in your tables.[3 marks]2 The Central Limit TheoremFor a sequence X1, X2, . . . , Xn of i.i.d. random variables with finiteexpectation E(Xi) = µ and variance Var(Xi) = σ2 > 0 for all 1 ≤ i ≤ n, wedefined the sample meanXn =X1 + X2 + . . . + Xnn.You can use the runif function in R to generate a Uniform(0, 1) random variable and use this to simulatethe random variable X which records the result of tossing a fair six-sided die.(Hint: Multiply by 6 and round up!)Forn = 1, 5, 10, 20, 30, 50, 100, 1000,generate a n × 1000-array of (independent) random variables, each with thedistribution of X.Recall that the Central Limit theorem states that for large enough n, we haveX ∼ N(µ, (σ^2)/n).For each array you generated above calculate the 1000 values of Xn for each n,and plot an appropriate histogram of the values. Overlay your plot with thenormal curve giving the density function of N(µ, σ2/n). Use built-in functionsin R to plot your diagram.