In non-examinable sections Section 13.4 and 13.5 I have tried to show you some moderately sophisticated applications of the simple ideas we are learning about p.g.f. and conditional expectation. I have added an Appendix B to the notes to say a bit more about the fast Fourier transform.

The brief story is this. To find the distribution of $Y$, the sum of two i.i.d. r.vs $X_1$ and $X_2$ taking values in $\{0,1,\dotsc,N-1\}$, we might find its p.g.f. $p_Y(z)=p(z)^2=(p_0+p_1z+\cdots+p_{N-1}z^{N-1})^2$.

This would take $O(N^2)$ multiplications of the form $p_ip_j$.

However, it is quicker to evaluate $p(z)$ at $2N$ complex values $z=\omega^0,\dotsc,\omega^{2N-1}$, square those values, and then recover the real coefficients of the polynomial $p(z)^2$. The calculation simplifies because by taking $\omega$ as a $2N$th root of unity the calculation of the $p(\omega^k)$, $k=0,\dotsc,2N-1$, can be done by a clever method, the discrete fast Fourier transform, which needs only $O(N\log N)$ multiplications.

Further details are in the notes on pages 52, 53 and 99, and then in the Part II course, Numerical Analysis.

The brief story is this. To find the distribution of $Y$, the sum of two i.i.d. r.vs $X_1$ and $X_2$ taking values in $\{0,1,\dotsc,N-1\}$, we might find its p.g.f. $p_Y(z)=p(z)^2=(p_0+p_1z+\cdots+p_{N-1}z^{N-1})^2$.

This would take $O(N^2)$ multiplications of the form $p_ip_j$.

However, it is quicker to evaluate $p(z)$ at $2N$ complex values $z=\omega^0,\dotsc,\omega^{2N-1}$, square those values, and then recover the real coefficients of the polynomial $p(z)^2$. The calculation simplifies because by taking $\omega$ as a $2N$th root of unity the calculation of the $p(\omega^k)$, $k=0,\dotsc,2N-1$, can be done by a clever method, the discrete fast Fourier transform, which needs only $O(N\log N)$ multiplications.

Further details are in the notes on pages 52, 53 and 99, and then in the Part II course, Numerical Analysis.