# Joint density of transformed independent uniform random variables

I'm taking a course in probability and struggling with transformations of random variables. An example in the text is:

Let U1 and U2 be independent uniform random variables, both on (0,1). Let Y1 = U1U2 and let Y2=U2.

(a) Find the joint density of Y1 and Y2.

If I calculate the inverse functions for U1 and U2, I get:

U1 = Y1/Y2

U2 = Y2

But I feel like I'm already on shaky ground because Y1 and Y2 defined over (0,1), which makes U1 undefined at Y2=0. Regardless, calculating the Jacobian gives my 1/Y2 - again, feels like shaky ground.

From this, the joint density should be 1/Y2 since this is defined as the product of the densities of U1 and U2 (uniform over (0,1), so densities equal 1) and the Jacobian is 1/Y2.

(b) What is the marginal density for Y1?

If I take the integral from 0 to 1 of the density above, I get -infinity. The integral of 1/Y2 is log(Y2), and I have to find that at 0 for the lower limit of the integral.

This seems like it should be an easy problem, but I can't wrap my head around where I've gone wrong.

Here is a rough outline to compute the joint:

$$P(Y_1 \le y_1, Y_2 \le y_2)=P(U_1U_2\le y_1, U_2\le y_2)$$

The right hand side can be re-written as:

$$P(U_1U_2\le y_1 | U_2\le y_2)\: P( U_2\le y_2)=P(U_1U_2\le y_1 | U_2\le y_2) \:y_2$$

Now, $P(U_1U_2\le y_1 | U_2\le y_2)$ can be written as:

$$P(U_1U_2\le y_1 | U_2\le y_2) = \int_0^{y_2}P(U_1 \le \frac{y_1}{y}) \: dy$$ If $y_2 \ge y_1$ then the integral simplifies to:

$$\int_0^{y_1} dy + \int_{y_1}^{y_2}\frac{y_1}{y} \: dy = y_1 + y_1[\log(y_2)-\log(y_1)]$$

If $y_2 < y_1$ then the integral simplifies to:

$$\int_0^{y_1} dy = y_1$$

Thus, the joint can be written as:

$$y_1 \cdot y_2 I(y_2<y_1) + y_2(y_1+y_1[\log(y_2)-\log(y_1)]) \: I(y_2\ge y_1)$$

The above can be simplified to:

$$y_1 \cdot y_2 \left\{ 1 + (\log(y_2)-\log(y_1))\: I(y_2\ge y_1) \right\}$$

I hope the above helps you start on the right track.

• I think I see what you did - converted the probability to a conditional probability and recognized that Y1 <= Y2. Can you recommend an online tutorial for these types of problems or a good probability text? The text I have is written by the instructor and lacks explanation plus had many errors. – KirkD_CO Oct 10 '17 at 12:57
• Starting with the joint CDF, using the conditional probability formula to simplify and then proceed further down that path is a standard approach. Recognizing how the integral depends on whether $y_2 > y_1$ or not requires some careful thinking. In general, in problems such as this it helps to be a bit careful about limits of integration as they are often a bit subtle. – Anon Oct 10 '17 at 13:02
• I see that now. Thanks for the post and explanation. My text skips the explanation through the conditional and none of the worked examples require a change to the limits of integration. A better text or resource would help. – KirkD_CO Oct 10 '17 at 13:05
• Working through your answer, I see in the first integral the appearance of y with no subscript, sp. Y1/Y. Was that done just to have a placeholder for Y2 which is effectively provided by the integration limits? – KirkD_CO Oct 10 '17 at 13:13
• In that integral, $y$ is just a dummy integration variable. You can replace that with some other letter, say $x$, with the limits still going from $0$ to $y_2$. So, it would be incorrect to say that it is a placeholder for $y_2$. In fact, if anything it is a placeholder for $U_2$ as we are integrating over the possible values for $U_2$ as it ranges from $0$ to $y_2$. – Anon Oct 10 '17 at 13:25