Logistic

dist=LogisticDistribution[μ,β]

PDF[dist,z]

CDF[dist,z]

Manipulate[Plot[PDF[LogisticDistribution[μ,β],x],{x,-10,10},Filling->Axis],{μ,-2,3},{β,1,3}]

Manipulate[Plot[Table[PDF[LogisticDistribution[μ,β],x],{μ,{-2,0,3}}]//Evaluate,{x,-10,10},Filling->Axis],{β,1,3}]

Manipulate[Plot[Evaluate[Table[PDF[LogisticDistribution[μ,β],x],{β,{1,2,3}}]],{x,-8,10},Filling->Axis],{μ,-2,3}]

Mean[dist]

Median[dist]

Variance[dist]

StandardDeviation[dist]

Skewness[dist]

Kurtosis[dist]

CharacteristicFunction[dist,t]

Random[LogisticDistribution[1,1]]