Finding the joint CDF using the joint PDF; why can't I do this?

Question

Find the joint CDF of the independent random variables $X$ and $Y$, where

$f_X(x)=x/2, 0\le x \le 2, $ and

$f_Y(y)=2y, 0 \le y \le 1$.

To do this, we can find the CDF separately for each of the marginal PDFs, and then multiply them together to get the joint CDF (since the variables are independent).

However, why doesn't it work to first multiply the marginal PDFs together, to get a joint PDF of $f_{X,Y}(x,y)=xy$, and then take the double integral (with respect to $x$ and then $y$) to get the joint CDF?

This is the way I thought to do it first but it seems like it wouldn't give the same answer; why not?

Thank you!

Answer 1

If $X$ and $Y$ are independent random variables, then $$F_{X,Y}(x,y)=\int_{-\infty}^x \int_{-\infty}^y f_{X,Y}(w,v)\,dv\,dw = \int_{-\infty}^x \int_{- \infty}^{y} f_X(w)f_Y(v)\,dv\,dw$$ $$=\int_{-\infty}^x f_X(w)\,dw\int_{-\infty}^{y}f_Y(v)\,dv = F_X(x)F_Y(y).$$

Method 1 (joint pdf approach) gives: $$f_{X,Y}(x,y)=f_X(x)f_Y(y)=xy,$$ if $0\leq x \leq 2, 0\leq y \leq 1$ and zero otherwise. Then $$F_{X,Y}(x,y)=\int_0^x \int_0^yf_{X,Y}(w,v)\,dv\,dw=\int_0^x \int_0^y wv \, dv \, dw=\frac{x^2y^2}{4}.$$

Method 2 (marginal cdf approach) gives: $$F_X(x) = \int_0^x \frac{w}{2} \, dw =\frac{x^2}{4}$$ $$F_Y(y) = \int_0^y 2v \, dv =y^2.$$ It follows $$F_{X,Y}(x,y)=F_X(x)F_Y(y)=\frac{x^2y^2}{4}.$$

I should note the cdf's are also piecewise functions: $F_X(x)$ and $F_Y(y)$ take a value of zero if $x \leq 0$ and $y \leq 0$, respectively, and take a value of 1 if $2 \leq x$ and $1 \leq y$, respectively. $F_{X,Y}(x,y)$ takes a value of zero if either $x \leq 0$ or $y \leq 0$, and takes a value of $1$ if $2 \leq x$ and $1 \leq y$.