How to reparametrize a probability density function

Question

I have this probability density function (pdf) $$ f(t)=\beta \left[\left(\frac{t}{\eta_1}\right)^{\beta} + \left(\frac{t} {\eta_2}\right)^{\beta} \right]t^{-1}e^{- \left(\frac{t}{\eta_1} \right)^{\beta} - \left(\frac{t}{\eta_2}\right)^{\beta}} $$ then using Mathematica 10 (student version), I obtain $$ \int_0^\infty f(t) dt= 1. $$ But when I reparametrize $\alpha_i = \left(\frac{t}{\eta_i}\right)^{\beta}$, $i=1,2$, $$ g(t) = \beta (\alpha_1 + \alpha_2)t^{-1}e^{- (\alpha_1 + \alpha_2)}, $$ I obtain an error warning:

Integrate::idiv: "Integral of 1/t does not converge on {0,∞}.

pdf:

model[t_, β_, η1_, η2_] := \
β  t^-1   ((t/η1)^β + (t/η2)^β) E^(-(t/\
η1)^β - (t/η2)^β)

Integrate[model[t, β, η1, η2], {t, 0, ∞}, 
 Assumptions -> {β > 0, η1 > 0, η2 > 0}]

latter code is equal 1.

pdf reparametrized:

model2[t_, β_, α1_, α2_] := β  t^-1   (α1 + α1) E^(- α1 - α2)

and when

Integrate[model2[t, β, α1, α2], {t, 0, ∞}, 
Assumptions -> {β > 0, α1 > 0, α2 > 0}]

I obtain this error warning:

Integrate::idiv: "Integral of 1/t does not converge on {0,∞}.

Answer 1

You have to make the substitutions back into the integrand, so Mathematica can know what the definitions of α1 and α2 are.

model2[t_, β_, α1_, α2_] := β(α1 + α2)/t E^(-α1 - α2)
   Integrate[
     model2[t, β, α1 /. α1 -> (t/n1)^β, α2 /. α2 -> (t/n2)^β], {t, 0, ∞}, 
     Assumptions -> {β > 0, n1 > 0, n2 > 0}]

1