# PDF of product of two continous joint distribution

Suppose that $X1$ and $X2$ have a continuous joint distribution for which the joint PDF is as follows:

\begin{equation*} f(x_1,x_2) = \begin{cases} x_1 + x_2 & \text {for $0 < x_1 < 1,$ and $0 < x_2 < 1$}\\ 0 & \text {otherwise}\\ \end{cases} \end{equation*}

Find the PDF of $Y = X_1 X_2$.

• Looked up wikipedia? – Karl Feb 1 '15 at 14:24
• Can you please guide me through the steps ? – mike111 Feb 1 '15 at 15:21
• What are the definitions and what haev you tried? – Alec Teal Feb 2 '15 at 10:35

Let's do it. Let $$F_Y$$ the CDF of $$Y$$.

$$F_Y(t) = P(Y \leq t) = P \left(X_1 \, X_2 \leq t \right) = P \left(X_1 \leq \dfrac{t}{X_2} \right).$$

The question here is: when does the inequality happen?

$$X_1 \leq \dfrac{t}{X_2}, \quad \text{when?}$$

For that, we need to find the marginal PDF of $$X_1$$ and $$X_2$$. How? With their joint distribution.

$$f_{X_1}(x_1) = \int_0^1 (x_1 + x_2)\, dx_2 = 1/2 + x_1$$ $$f_{X_2}(x_2) = \int_0^1 (x_1 + x_2)\, dx_1 = 1/2 + x_2$$

Ok, now... when does this happen?

$$1/2 + x_1 \leq \dfrac{t}{1/2 + x_2}$$

With $$x_1 = x_2$$, find $$x_1$$ (or $$x_2$$) in terms of $$t$$.

$$x_1 = 1/2 \cdot (-1 + 2 \sqrt t). \tag{1} \label{exacto}$$

If you draw $$1/2 \, + x_1$$ and $$t /( 1/2 + x_2)$$ between $$0$$ and $$1$$ (do it!), you see that the inequality is true for some values of $$t$$. So the CDF of $$Y$$ is:

$$F_Y(t) = \int_0^1 \left( \int_0^{1/2 \, (-1+ 2\sqrt t)} (x_1 + x_2) \, dx_1 \right) dx_2 = \dfrac{1}{8} (4 t-1).$$

We need to find an interval for $$t$$. What is it? Is $$t$$ in $$[0,1]$$ or in $$[1, 10]$$?. Refer to $$\eqref{exacto}$$ to find $$t$$ when $$x_1=0$$ and $$x_1=1$$, so $$t \in [1/4, 9/4]$$. The PDF is:

$$f_Y(t) = \dfrac{d}{dt}F_Y (t) = \dfrac{1}{2}.$$

Check that:

$$\int_{1/4}^{9/4} \dfrac{1}{2}\, dt = 1.$$