The two random variable X and Y are uncorrelated if and only if their covariance is

Question

Question

This question was previously asked in

ESE Electronics Prelims 2021 Official Paper

Answer 1

Two random variables are said to be uncorrelated if their covariance is zero.

Now,

$C o v (X, Y) = E [(X - μ_{X}) (Y - μ_{Y})]$

where $μ_{X} = E (X)$ and $μ_{Y} = E (Y)$

Thus, $C o v (X, Y) = E [X Y - μ_{X} Y - μ_{Y} X + μ_{X} μ_{Y}]$

$= E [X Y] - E [μ_{X} Y] - E [μ_{Y} X] + E [μ_{X} μ_{Y}]$

Using, the property of expectation that $E [c Y] = c E [Y]$ we have,

$\begin{array}{l} C o v (X, Y) = E (X Y) - μ_{X} μ_{Y} - μ_{Y} μ_{X} + μ_{X} μ_{Y} \\ C o v (X, Y) = E (X Y) - E (X) E (Y) \end{array}$

For uncorrelated variables, $C o v (X, Y) = 0$

Thus, for variables to be uncorrelated

E (X Y) = E (X) E (Y)