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

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

Now,

\( {\rm{Cov}}\left( {{\rm{X}},{\rm{Y}}} \right) = {\rm{E}}\left[ {\left( {{\rm{X}} - {{\rm{\mu }}_{\rm{X}}}} \right)\left( {{\rm{Y}} - {{\rm{\mu }}_{\rm{Y}}}} \right)} \right]\)

where \({{\rm{\mu }}_{\rm{X}}} = {\rm{E}}\left( {\rm{X}} \right)\) and \({{\rm{\mu }}_{\rm{Y}}} = {\rm{E}}\left( {\rm{Y}} \right)\)

Thus, \({\rm{Cov}}\left( {{\rm{X}},{\rm{Y}}} \right) = {\rm{E}}\left[ {{\rm{XY}} - {{\rm{\mu }}_{\rm{X}}}{\rm{Y}}-{{\rm{\mu }}_{\rm{Y}}}{\rm{X}} + {{\rm{\mu }}_{\rm{X}}}{{\rm{\mu }}_{\rm{Y}}}} \right] \) 

\(= {\rm{E}}\left[ {{\rm{XY}}} \right] - {\rm{E}}\left[ {{{\rm{\mu }}_{\rm{X}}}{\rm{Y}}} \right] - {\rm{E}}\left[ {{{\rm{\mu }}_{\rm{Y}}}{\rm{X}}} \right] + {\rm{E}}\left[ {{{\rm{\mu }}_{\rm{X}}}{{\rm{\mu }}_{\rm{Y}}}} \right]\)

Using, the property of expectation that \({\rm{E}}\left[ {{\rm{cY}}} \right] = {\rm{cE}}\left[ {\rm{Y}} \right]\) we have,

\(\begin{array}{l} {\rm{Cov}}\left( {{\rm{X}},{\rm{Y}}} \right) = {\rm{E}}\left( {{\rm{XY}}} \right)-{{\rm{\mu }}_{\rm{X}}}{{\rm{\mu }}_{\rm{Y}}}-{{\rm{\mu }}_{\rm{Y}}}{{\rm{\mu }}_{\rm{X}}} + {{\rm{\mu }}_{\rm{X}}}{{\rm{\mu }}_{\rm{Y}}}\\ {\rm{Cov}}\left( {{\rm{X}},{\rm{Y}}} \right) = {\rm{E}}\left( {{\rm{XY}}} \right)-{\rm{E}}\left( {\rm{X}} \right){\rm{E}}\left( {\rm{Y}} \right) \end{array}\)

For uncorrelated variables, \({\rm{Cov}}\left( {{\rm{X}},{\rm{Y}}} \right) = 0\)