Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 205 / 3 / b / Solution

Let $U$ be uniform on $[0,2\pi ]$, and independently let $W$ have densityThe supplied Fourier transform identity, after the change of Fourier convention, givesAlso ${\mathbb{E}}_U[\cos (a+U)\cos (b+U)]=\frac{1}{2}\cos (a-b)$. Thereforeso one may take $c=1$.