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Past exam of the mathematics course of the University of Oxford

Undated

Prelims

Mathematics III resit

{c}

For <random variables> with finite <second moments>, their <covariance> is
$$
\boxed{\operatorname{Cov}(X,Y)=\mathbb E[(X-\mathbb EX)(Y-\mathbb EY)].}
$$
The <Cauchy-Schwarz inequality> ensures the product is <integrable>. Expand and use <linearity of expectation>:
$$
\begin{aligned}
\operatorname{Cov}(X,Y)
&=\mathbb E[XY]-\mathbb E[X]\mathbb E[Y]
-\mathbb E[X]\mathbb E[Y]+\mathbb E[X]\mathbb E[Y]\\
&=\boxed{\mathbb E[XY]-\mathbb E[X]\mathbb E[Y].}
\end{aligned}
$$
The same identity holds whenever the quantities involved exist with sufficient <integrability>.

Solution

Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 5 / b / Solution

Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 5 / b / Solution

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Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 5 / b / Solution

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