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Natural sciences course of the University of Cambridge

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Year 1 of the natural sciences course of the University of Cambridge

Mathematics course of the year 1 of the natural sciences course of the University of Cambridge

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge

2012

Paper 1

When the relevant <moments> are finite, the <mean> and <variance> are
$$
\boxed{\mu=\mathbb E[X]=\int_{-\infty}^{\infty}x f(x)\,dx,\qquad
\sigma^2=\mathbb E[(X-\mu)^2]=\int_{-\infty}^{\infty}(x-\mu)^2f(x)\,dx.}
$$
Equivalently, $\sigma^2=\int x^2f(x)\,dx-\mu^2$. The <mean> requires an integrable absolute first <moment>; finite second <moment> guarantees both quantities are finite.

Solution

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 18 / b / ii / Solution

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 18 / b / ii / Solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 18 / b / ii / Solution

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