Determining independence of random variables from joint pdf

Question

I am working on the problem below and have a question about the note that is provided.

I know that independent random variables must satisfy the following;
$$
f_{XY}(x,y) = f_{_X}(x)f_{_Y}(y)
$$
where $f_{_X}(x)$ and $f_{_Y}(y)$ are the marginal PDFs of the joint density $f_{XY}(x,y)$

From this fact we can make the following observations;

i.
$$
f_{_X}(x) = \int\limits_{0}^{1} (2x + y)dy = 2x+0.5, ~~~~~~
f_{_Y}(y) = \int\limits_{0}^{1} (2x + y)dx = y+1
$$
Since $f_{XY}(x,y) \neq f_{_X}(x)f_{_Y}(y) $ we conclude that $X$ and $Y$ are not independent.

ii.
$$
f_{_X}(x) = \int\limits_{0}^{1} 2 \lambda e^{-\lambda x} y ~ dy = \lambda e^{-\lambda x}, ~~~~~~
f_{_Y}(y) = \int\limits_{0}^{\infty} 2 \lambda e^{- \lambda x} y ~ dx = 2y
$$
Since $f_{XY}(x,y) = f_{_X}(x)f_{_Y}(y) $ we conclude that $X$ and $Y$ are independent.

iii.
$$
f_{_X}(x) = \int\limits_{0}^{\sqrt{2}} 2 x y ~ dy = 2x, ~~~~~~
f_{_Y}(y) = \int\limits_{0}^{1} 2 x y ~ dx = y
$$
Since $f_{XY}(x,y) = f_{_X}(x)f_{_Y}(y) $ we conclude that $X$ and $Y$ are independent.

I know how to do the problem if I have the marginal PDFs; however, I haven't a clue as to how to determine independence if I don't. The note indicates that it is possible.

Answer 1

Consider the following result:

Let $X$ and $Y$ random variables with joint density $f(x,y)$ positive if and only if $(x,y)\in [a,b]\times [c,d]$ and $f(x,y)=0$ otherwise. Then, $$X\perp Y\iff f(x,y)=g(x)h(y),$$where $\perp$ denote independence between random variables and $g$ is non-negative function of $x$ and $h$ is non-negative function of $y$.

The result above indicates that we do not need under certain conditions to find the marginal densities, as the given problem indicates.

For example, notice we have

• $f(x,y)=2xy$ is positive for $(x,y)\in [0,1]\times [0,\sqrt{2}]$ and $f(x,y)=0$ in otherwise. Also, $f(x,y)=g(x)h(y)$ where

$$g(x)=\begin{cases}x,\quad 0\leqslant x\leqslant 1,\\0,\quad \text{otherwise}\end{cases}, \quad h(y)=\begin{cases}2y,\quad 0\leqslant y\leqslant \sqrt{2},\\0,\quad \text{otherwise}\end{cases}$$

Thus, $X$ and $Y$ are independent random variables.