could you produce 3 histogram ggplots for this question, similar to the ones i uploaded using RPlot the Histogram for the Exponential distributionPlot the Histogram for the Normal distributionplot for comparison between exponential and normal,with a key and also could you outline the graphs with a normal curve and provide an explanation for th eobservations and what the graph tells us . 04IDemonstrating Central Limit Theorem1.ValueThe Distribution of Sample Means and the Uniform Distribution04Distribution of MeansUno Distribution00ampla ass trearotidal variazas 14 0.001611049547[1] "The hmber of similations in 2008# [1] "The difference between sample and theoretical means is 0.00151247649783848"# [1] "The difference between sample and theoretical variances is 0.0012342020050992"Demonstrating Central Limit TheoremUniform Distribution1500-ValueValueTypeHulhNormal DibutionThe Distribution of Sample Means and the Uniform DistributionValueType[1] "The Number of simulations is 10000"[1] "the difference between sample and theoretical means is 0.000366877283006026"# [1] "The difference between sample and theoretical variances is 0.008186210166330476Demonstrating Central Limit Theoremof Sample Manbution of Sample ManNoutonValueThe Distribution of Sample Means and the Uniform DistributionDistribution of MeansUniform DistributionOtribution of Sample MeanNormal Dibution The CLT states that, for large n, the distribution of the sample mean approaches a Normal distribution.Mathematically,X₁ +" N (1, 0-²)9nXnwhere →→D means 'converges in distribution'; it's implied here that this convergence takes place as ŉ, or thenumber of underlying random variables, grows.This is an extremely powerful result, because it holds no matter what the distribution of the underlying randomvariables is.Task Using software R, demonstrate that CLT holds on the example the Exponential Distribution. Use λ = 5 forde-2x, x ≥ 0.

Algorithm:Set the seed for reproducibility.Generate random samples from the Exponential distribution…

The CLT states that, for large n, the distribution of the sample mean approaches a Normal distribution. Mathematically, Xn X₂ →º N (µ‚, 2/² ). D n where →D means 'converges in distribution'; it's implied here that this convergence takes place as n, or the number of underlying random variables, grows. This is an extremely powerful result, because it holds no matter what the distribution of the underlying random variables is. Task Using software R, demonstrate that CLT holds on the example of the Exponential Distribution. Use λ = 5 for de-ix ,x ≥ 0.

The CLT states that, for large n, the distribution of the sample mean approaches a Normal distribution. Mathematically, Xn X₂ →º N (µ‚, 2/² ). D n where →D means 'converges in distribution'; it's implied here that this convergence takes place as n, or the number of underlying random variables, grows. This is an extremely powerful result, because it holds no matter what the distribution of the underlying random variables is. Task Using software R, demonstrate that CLT holds on the example of the Exponential Distribution. Use λ = 5 for de-ix ,x ≥ 0.

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter12: Review Of Calculus And Probability

Section12.5: Random Variables, Mean, Variance, And Covariance

Problem 5P

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