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Year 2 of the mathematics course of the University of Oxford

Probability course of the University of Oxford

Past exam of the probability course of the University of Oxford

2014

By part (c) with $\lambda=1$, let $T_n=E_1+\cdots+E_n\sim\operatorname{Gamma}(n,1)$ for <independent and identically distributed> <exponential random variables>. The requested integral is $\mathbb P(T_n\leq n)$. Each summand has <expectation> one and <variance> one, so the <central limit theorem> gives
$$
\frac{T_n-n}{\sqrt n}\xrightarrow{d}N(0,1).
$$
Zero is a <continuity> point of the <standard normal distribution>, and consequently
$$
\boxed{\lim_{n\to\infty}\frac1{(n-1)!}\int_0^n x^{n-1}e^{-x}\,dx=\Phi(0)=\frac12.}
$$

Solution

Past exam of the probability course of the University of Oxford / 2014 / 2 / e / Solution

Past exam of the probability course of the University of Oxford / 2014 / 2 / e / Solution

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