Laplace

dist = LaplaceDistribution[μ,β]

PDF[dist,z]

CDF[dist,z]

Manipulate[Plot[PDF[LaplaceDistribution[μ,β],x],{x,0,5},Filling->Axis,PlotRange->All,Exclusions->None],{μ,2,4},{β,0.5,2}]

Manipulate[Plot[Table[PDF[LaplaceDistribution[μ,β],x],{β,{0.5,1,2}}]//Evaluate,{x,0,5},Filling->Axis,PlotRange->All,Exclusions->None],{μ,2,4}]

Manipulate[Plot[Table[PDF[LaplaceDistribution[μ,β],x],{μ,{2,3,4}}]//Evaluate,{x,0,5},Filling->Axis,PlotRange->All,Exclusions->None],{β,0.5,2}]

Mean[dist]

Median[dist]

Variance[dist]

StandardDeviation[dist]

Skewness[dist]

Kurtosis[dist]

CharacteristicFunction[dist,t]

Random[LaplaceDistribution[1,1]]