Expectation of Product of Independent Random Variables is Product of Expectations/Corollary

Corollary to Expectation of Product of Independent Random Variables is Product of Expectations

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ and $Y$ be integrable random variables that are independent.

Then:

$\expect {X Y} = \expect X \expect Y$

Proof

We have:

\(\ds \expect {X Y}\)

\(=\)

\(\ds \expect {\paren {X^+ - X^-} \paren {Y^+ - Y^-} }\)

\(\ds \)

\(=\)

\(\ds \expect {X^+ Y^+ - X^+ Y^- - X^- Y^+ + X^- Y^-}\)

\(\ds \)

\(=\)

\(\ds \expect {X^+ Y^+} - \expect {X^+ Y^-} - \expect {X^- Y^+} + \expect {X^- Y^-}\)

Expectation is Linear

\(\ds \)

\(=\)

\(\ds \expect {X^+} \expect {Y^+} - \expect {X^+} \expect {Y^-} - \expect {X^-} \expect {Y^+} + \expect {X^- Y^-}\)

Expectation of Product of Independent Random Variables is Product of Expectations

\(\ds \)

\(=\)

\(\ds \expect {X^+} \paren {\expect {Y^+} - \expect {Y^-} } - \expect {X^-} \paren {\expect {Y^+} - \expect {Y^-} }\)

\(\ds \)

\(=\)

\(\ds \expect {X^+} \expect {Y^+ - Y^-} - \expect {X^-} \expect {Y^+ - Y^-}\)

Expectation is Linear

\(\ds \)

\(=\)

\(\ds \expect {X^+} \expect Y - \expect {X^-} \expect Y\)

\(\ds \)

\(=\)

\(\ds \expect Y \expect {X^+ - X^-}\)

Expectation is Linear

\(\ds \)

\(=\)

\(\ds \expect X \expect Y\)

$\blacksquare$