Question

Two random variables $X$ and $Y$ have marginal probability density functions (PDFs) $f_X\left(x\right)$ and $f_Y\left(y\right)$ respectively, and they have a joint PDF of $f_{XY}(x,y)$. If $E[.]$ represents the expectation operator, then consider the following relations:
1. $E\left[XY\right]=E\left[X\right]E\left[Y\right]$
2. $f_{XY}(x,y)=f_X\left(x\right)f_Y\left(y\right)$
Which of the above two relations can confirm the statistical independency of $X$ and $Y$?

• A. Neither $1$ and $2$
• B. Both $1$ and $2$
• C. $2$ only
• D. $1$ only

Answer 1

The correct option is C $2$ only

Only $f_{XY}(x,y)=f_X\left(x\right)f_Y\left(y\right)$ confirms the independency of $X$ and $Y$.

If $X$ and $Y$ are independent, then $E\left[XY\right]=E\left[X\right]E\left[Y\right]$; but converse may not be true always. For example, let $X=cos\theta$ and $Y=sin\theta$, where $\theta$ is uniformly distributed over $[0,2\pi ]$. For this variable $X$ and $Y$, $E\left[XY\right]=E\left[X\right]E\left[Y\right]=0$ relation is satisfied, but they are not independent. It can be confirmed from $E\left[X^2\right]=1/2,E\left[Y^2\right]=1/2$ and $E\left[X^2{Y}^2\right]=1/8\ne E\left[X^2\right]E\left[Y^2\right]$. So $X$ and $Y$ are not independent but $E\left[XY\right]=E\left[X\right]E\left[Y\right]$ is satisfied.