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Year 2 of the mathematics course of the University of Oxford

Options II course of the University of Oxford

Past exam of the options II course of the University of Oxford

2011

Differentiate the <probability generating function>:
$$
g'(z)=\frac52(14-5z)^{-3/2},\qquad
g''(z)=\frac{75}4(14-5z)^{-5/2}.
$$
Therefore
$$
\mathbb ET=g'(1)=\frac5{54},\qquad
\mathbb E[T(T-1)]=g''(1)=\frac{25}{324}.
$$
Using the relation between the second factorial moment and <variance>,
$$
\boxed{\operatorname{Var}(T)=g''(1)+g'(1)-g'(1)^2
=\frac{235}{1458}.}
$$

Solution

Past exam of the options II course of the University of Oxford / 2011 / M / M4 / b / ii / Solution

Past exam of the options II course of the University of Oxford / 2011 / M / M4 / b / ii / Solution

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