Question

$X$ and $Y$ are two statistically independent random variables. If $X$ has uniform PDF in the interval $[0,2]$ and $Y$ has uniform PDF in the interval $[0,4]$, then the value of $E\left[XY\right]$ will be

• A. $3$
• B. $2$
• C. $0$
• D. $1$

Answer 1

The correct option is B $2$

As $X$ and $Y$ statiscally undependent,

$E\left[XY\right]=E\left[X\right]E\left[Y\right]$
$E\left[X\right]={\int }_{-\infty }^{\infty }x{f}_X\left(x\right)dx=1\because f_X\left(x\right)issymmetricaboutthepointx=1$
$E\left[Y\right]={\int }_{-\infty }^{\infty }y{f}_Y\left(y\right)=2\because f_Y\left(y\right)issymmetricaboutthepointy=2$

so, $E\left[XY\right]=E\left[X\right]E\left[Y\right]=\left(1\right)\left(2\right)=2$