Characteristic function of the uniform random variable on $[0,1]^2$

Question

I need to calculate the characteristic function of the uniform variable on $[0,1]^2$.

My idea was the following:

Let $(X,Y)\sim \mathrm{Unif}\left([0,1]^2\right)$. First remark that $f_{(X,Y)}(x,y)=1_{[0,1]^2}(x,y)=1_{[0,1]}(x)1_{[0,1]}(y)=f_X(x)f_Y(y)$ hence $X$ is independent from $Y$. Now $$\Phi_{(X,Y)}(u,v)=\Bbb{E}\left(e^{i\langle(u,v),(X,Y)\rangle}\right)=\Bbb{E}\left(e^{iuX}e^{ivY}\right)\stackrel{\text{independent}}{=}\Bbb{E}\left(e^{iuX}\right)\Bbb{E}\left(e^{ivY}\right)=\frac{e^{iu}-1}{iu}\cdot \frac{e^{iv}-1}{iv}. $$

Is this correct so or can't I do this like this?

Thanks for your help.

Answer 1

What you did is correct. The key point was indeed to use independence between $X$ and $Y$.

Maybe you should precise what happens when $u$ or $v$ equals $0$ [there is no problem as you can take the limit as $u,v\to 0$].