The coordinate-deletion functions from part c show that $Z$ is weakly self-bounding. Its negative centered moment-generating function therefore satisfies the <lower-tail concentration for a weakly self-bounding function> estimate
$$
\log\mathbb E e^{-\lambda(Z-\mathbb EZ)}
\leq\frac{\lambda^2\mathbb EZ}{2},
\qquad \lambda\geq0.
$$
Applying the <Chernoff bound>, for every $\lambda\geq0$,
$$
\mathbb P(Z-\mathbb EZ<-t)
\leq\exp\left(-\lambda t+\frac{\lambda^2\mathbb EZ}{2}\right).
$$
The exponent is minimized at $\lambda=t/\mathbb EZ$, giving
$$
\mathbb P(Z-\mathbb EZ<-t)
\leq\exp\left(-\frac{t^2}{2\mathbb EZ}\right).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 208 / 4 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-208/4/d/solution