Central Limit Theorem

Uniform Distribution

Manipulate[
SeedRandom[seed];
Show[
Histogram[Table[Mean[RandomReal[{0,100}, n]], {200}], {35, 65, 1}],
Plot[200*PDF[NormalDistribution[50, Sqrt[9999/(12n)]],x], {x,35,65}],
ImageSize->{500, 300}
] ,  
{{n,20,"sample size"}, 10,200,1, Appearance->"Labeled"}, 
{{seed,1,"new sample"},1,100000,1}]

Bernoulli Distribution

Manipulate[
SeedRandom[seed];
Show[
Histogram[Table[Mean[RandomVariate[BernoulliDistribution[0.5],n]], {5000}], {0.2, 0.8, 0.001}],
Plot[25*PDF[NormalDistribution[0.5, Sqrt[0.25/n]],x], {x,0.2,0.8}],
ImageSize->{500, 300}
] ,  
{{n,20,"sample size"}, 10,200,1, Appearance->"Labeled"}, 
{{seed,1,"new sample"},1,100000,1}]

Exponential Distribution

Manipulate[
SeedRandom[seed];
Show[
Histogram[Table[Mean[RandomVariate[ExponentialDistribution[0.2],n]], {500}], {2, 8, 0.1}],
Plot[50*PDF[NormalDistribution[5, Sqrt[25/n]],x], {x,2,8}],
ImageSize->{500, 300}
] ,  
{{n,20,"sample size"}, 10,200,1, Appearance->"Labeled"}, 
{{seed,1,"new sample"},1,100000,1}]

expMeans=Mean/@Table[RandomVariate[ExponentialDistribution[1],20],20]
{Mean[expMeans],StandardDeviation[expMeans]}