# Change of variables in pdf

I have the joint pdf$$f(x_1,x_2)=x_1e^{-x_1(1+x_2)}I_{(0,\infty)}(x_1)I_{(0,\infty)}(x_2)$$and have to derive the joint pdf of $$Y_1=e^{-X_1}\qquad\text{ and }\quad Y_2=e^{-X_1X_2}$$ I set $$x_1=-\ln(y_1)$$ and $$x_2=\ln(y_2)/\ln(y_1)$$. When I plug these transforms into $$f(x_1,x_2)$$ and multiply with the absolute determinant of the Jacobian $$|\det(J)|=1/(y_1y_2\ln(y_2))$$, I get a negative result. Where did I make a mistake?

• Can you share you calculations in detail? And, $\ln y_2$ can be negative, but you got it out of the absolute value expression. – gunes Jul 8 at 9:29
• The Jacobian should be $1/|y_1y_2\log(y_1)|$, I think. – Xi'an Jul 8 at 9:57
• You're right, I made a typo there. But when I plug that Jacobian along with $g^{-1}_1(y_1)=x_1=-\ln(y_1)$ and $g^{-1}_2(y_2)=x_2=\ln(y_2)/\ln(y_1)$ into $h(y_1,y_2)=f(g^{-1}_1(y_1),g^{-1}_2(y_2))|det(J)|$, I get $h(y1,y2)=-1$ - and that can't be right, or am I mistaken? – Niklas Jul 8 at 10:12
• You have to keep the absolute value around $\text{det}J$, meaning$$|\ln(y_1)|=-\ln(y_1)$$ – Xi'an Jul 8 at 10:41
• I'm sorry, I don't quite understand. Why do I have to form the absolute value of $-\ln(y_1)$? It's part of the pdf and not part of $|det(J)|$ – Niklas Jul 8 at 10:59

Let's first explore how much progress we can make without trying to solve for the x's in terms of the y's and by avoiding a direct calculation of the Jacobian (according to the Principle of Mathematical Laziness).

From

$$\mathrm{d}y_1 = -e^{-x_1}\mathrm{d}x_1$$

and

$$\mathrm{d}y_2 = -e^{-x_1x_2}\left(x_2\mathrm{d}x_1 + x_1\mathrm{d}x_2\right),$$

both computed using elementary rules of differentiation, notice that

$$\mathrm{d}y_1\wedge \mathrm{d}y_2 = \left(-e^{-x_1}\right)\left(-e^{-x_1x_2}\right)\left(x_1 \mathrm{d}x_1\wedge\mathrm{d}x_2\right) = x_1e^{-x_1(1+x_2)}\mathrm{d}x_1\wedge\mathrm{d}x_2,$$

which we may use in a first step towards transforming the probability element:

$$f_{X_1,X_2}(x_1,x_2)\mathrm{d}x_1\mathrm{d}x_2 = \mathcal{I}_{(0,\infty)}(x_1)\mathcal{I}_{(0,\infty)}(x_2)\,\mathrm{d}y_1\mathrm{d}y_2.\tag{*}$$

(This is a bit of an abuse of notation: we must think of the $$x_i$$ on the right hand side as being functions of the $$y_i,$$ whereas on the left hand side the $$x_i$$ are just variables.)

It remains only to re-express the indicator functions in terms of $$(y_1,y_2).$$ Since $$0 \lt x_1 \lt \infty,$$

$$1 = e^{-0} \gt e^{-x_1} = y_1 \gt e^{-\infty} = 0$$

and

$$1 = e^{-0} \gt e^{-x_1x_2} = y_2 \gt e^{-\infty(\infty)} = 0.$$

Thus $$(*)$$ becomes

$$f_{X_1,X_2}(x_1,x_2)\mathrm{d}x_1\mathrm{d}x_2 = \mathcal{I}_{(0,1)}(y_1)\mathcal{I}_{(0,1)}(y_2)\,\mathrm{d}y_1\mathrm{d}y_2$$

from which we can read off the density as

$$f_{Y_1,Y_2}(y_1,y_2) = \mathcal{I}_{(0,1)}(y_1)\mathcal{I}_{(0,1)}(y_2).$$

This is, of course, the uniform density on the unit square $$(0,1)^2.$$ As a check, let's plot some simulated values of $$(Y_1,Y_2).$$ In R this can be carried out as

n <- 1e4
x1 <- rexp(n)
x2 <- rexp(n, x1)
y1 <- exp(-x1)
y2 <- exp(-x1*x2)
plot(y1, y2, asp=1, xaxp=c(0, 1, 2), yaxp=c(0, 1, 2),
pch=19, cex=1/2, col="#00000010",
main=expression(group("(", list(Y, Y), ")")),
xlab=expression(y), ylab=expression(y))


(This works because $$X_1$$ has an exponential distribution and, conditional on $$X_1,$$ $$X_2$$ has an exponential distribution with rate $$X_1.$$) The plot of the y-values indeed fills the unit square uniformly (up to expected statistical fluctuations): 