# How to reparametrize a probability density function

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,∞}.

• Please use code, rather than $\LaTeX$. – Feyre Dec 15 '16 at 21:37
• Similar – corey979 Dec 15 '16 at 22:14
• @Feyre, Where am I wrong? I'm novice in to use Mathematica software. – Marco Dec 15 '16 at 22:30
• I do not get the error, in fact your second code evaluates to 1 too, I suggest quitting your kernel and trying again. – Feyre Dec 15 '16 at 22:44
• @Feyre There's an error in the second code: there's model while it should be model2. – corey979 Dec 15 '16 at 23:11

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)

1
• @Marco, "I can't have $\eta_i$ in my outcomes" - that's kind of muddled; you do realize that $\eta_i$ is constant, while $\alpha_i$ is dependent on $t$, no? Thus, your mean expression cannot involve $\alpha_i$ as you have presented it. – J. M. will be back soon Dec 16 '16 at 2:23