Clear[n]

fn[x_]:=PDF[NormalDistribution[μ,σ],x]

Li2[x_]=((∂_μ) Log[fn[x]])^2
Lf[x_]=Li2[x]*fn[x]

cr1=n*Integrate[Lf[x],{x,-∞,∞},Assumptions->{μ∈Reals,σ>0}]
cr=1/cr1

Li2[x_]=Simplify[(∂_σ) Log[fn[x]]]^2
Lf[x_]=Simplify[Li2[x]*fn[x]]

cr1=n*Integrate[Lf[x],{x,-∞,∞},Assumptions->{μ∈Reals,σ>0}]
cr=1/cr1

fe[x_]:=PDF[ExponentialDistribution[λ],x]

Li2[x_]=Simplify[((∂_λ) Log[fe[x]])^2,Assumptions->{λ>0}]
Lf[x_]=Simplify[Li2[x]*fe[x]]

cr1=n*Integrate[Lf[x],{x,0,∞},Assumptions->{λ>0}]
cr=1/cr1

fb[k_]:=PDF[BernoulliDistribution[p],k]

fb[k]

Li2[k_]=((∂_p) Log[fb[k]])^2
Lf[k_]=Simplify[Li2[k]*fb[k]]

cr1=n*Simplify[Sum[Lf[k],{k,0,1}],Assumptions->{p>0,p<1}]
cr=1/cr1

fp[x_]:=PDF[PoissonDistribution[λ],x]

Li2[x_]=Simplify[(∂_λ) Log[fp[x]],Assumptions->{λ>0,x≥0}]^2
Lf[x_]=Simplify[Li2[x]*fp[x]]

cr1=n*Simplify[Sum[Lf[k],{k,0,∞}],Assumptions->{λ>0}]
cr=1/cr1

Simplify[1/p+1/(1-p)]