Set $x_*=0.5$ in the <Gaussian process regression posterior> and compute $m_*$ and $v_*$ as in part a. <Standardization of a normal random variable> then gives
$$
\mathbb P\bigl(f(0.5)<1.5\mid y,X\bigr)
=\Phi\!\left(\frac{1.5-m_*}{\sqrt{v_*}}\right),
$$
where $\Phi$ is the <standard normal cumulative distribution function>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 3 / b / Solution

If $X\sim N(\mu,\sigma^2)$ with $\sigma>0$, then $(X-\mu)/\sigma$ has the <standard normal distribution>.

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 Incoming links: Standardization of a normal random variable