Showing the X and Y are Independent when their pdf is given

Question

I am supposed to show that X and Y are independent. I have my suspicions that maybe I should use some sort of ecpectations rule, but I am honestly unsure.

This is as far as I have gotten: $f_{X,Y}({x,y})=\{_{0\ \ \ o.w.}^{1 \ \ \ if \ \ 0<x<1 and 0<y<1}$

I wrote out the the marginal pdfs like so. ( I believe this is correct )

$f_X(x)=\int_{-\infty}^{\infty}f_{X,Y}(x,y)dy$

and

$f_Y(y)=\int_{-\infty}^{\infty}f_{X,Y}(x,y)dx$

I will also include a screenshot of the original question for more context.

Answer 1

We know the joint PDF of $[X,Y]$, so in this case we can calculate the marginal distributions with the formulas you have given above. (Perhaps there are some exceptions, but those are irrelevant now.)

The PDF of $X$ is $$f_{X}\left(x\right)=\begin{cases} 1 & \text{if}\;x\in\left[0,1\right]\\ 0 & \text{otherwise} \end{cases},$$

and the PDF of Y is: $$f_{Y}\left(y\right)=\begin{cases} 1 & \text{if}\;y\in\left[0,1\right]\\ 0 & \text{otherwise} \end{cases}.$$

While $$f_{X,Y}\left(x,y\right)=f_{X}\left(x\right)\cdot f_{Y}\left(y\right),$$ therefore $X$ and $Y$ are independent. (We can say $X$ and $Y$ are independent if their joint PDF can be calculated with the products of the two marginal PDF.)

Answer 2

You need to show that $$ f_{X,Y}(x,y) = f_X(x)f_Y(y) $$ for all $(x, y)$. The marginal pdf of $X$ is $$ f_X(x) = \int f_{X,Y}(x,y)dy = \int1 dy=1, $$ same for $f_Y(y)$, hence $$ f_X(x)f_Y(y) = 1\cdot 1 =1 = f_{X,Y}(x,y) $$ for $(x, y)$ in $[0,1]^2$, and $0$ otherwise.