<jupyter_start><jupyter_text>Central Limit Theorem for Different Distributions: It can be observed that for all the distribution types considering 100 distributions of sample size 100 each the weight of the graph is concentrated towards the Mean of the distribution. This justifies that all the distributions that we have plotted follow central limit theorem.\ Normal Distribution "The central limit theorem states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement , then the distribution of the sample means will be approximately normally distributed" It is very clearly evident that the normal distribution obeys central limit theorem. Poisson Distribution "Poisson(100) distribution can be thought of as the sum of 100 independent Poisson(1) variables and hence may be considered approximately Normal, by the central limit theorem, so Normal( μ = rate*Size = λ*N, σ =√(λ*N)) approximates Poisson(λ*N = 1*100 = 100)" Since Poisson can be approximated to a normal distribution. It obeys Central Limit Theorem. Binomial Distribution For the binomial distribution b(n, p, j) we have  limn→∞ √npq b(n, p,{np + x √npq}) = φ(x) So higher the distribution, induvidual probabilities turn close to zero making n tending to ∞ making it obey CLT. ChiSquare Distribution If there we take standardized normal distribtution as in our case, the distribution tends to follow the standard normal distribution Laplace Distribution Laplace Distribution clearly follows the Central Distribution Theorem.<jupyter_code># Plotting all the distributions together as a histogram
Normal = list(md['Normal']['means'].values())
Poisson = list(md['Poisson']['means'].values())
Binomial = list(md['Binomial']['means'].values())
ChiSquare = list(md['ChiSquare']['means'].values())
Laplace = list(md['Laplace']['means'].values())
pl.figure(figsize=(8,8))
pl.hist((Normal, Poisson, Binomial, ChiSquare, Laplace),bins=30,stacked=True)
pl.xlabel('Sample Mean', fontsize = 18)
pl.ylabel('No. of Samples', fontsize = 18)
pl.legend(('Chisquared','Normal','Poisson','Binomial','LaPlace'),loc='upper right',bbox_to_anchor=(1.25,1))
pl.title("All Distributions", fontsize = 18)<jupyter_output><empty_output>
StarcoderdataJupyterStructuredCleanDedup