The <chain rule for information entropy> and <conditioning reduces entropy> give
$$
H(X,Y)=H(X)+H(Y\mid X)\leq H(X)+H(Y).
$$
Thus <subadditivity of information entropy> holds for two finitely valued <random variables>. Applying this inequality to $(X_1,\ldots,X_{n-1})$ and $X_n$ and then inducting gives
$$
\boxed{H(X_1,\ldots,X_n)\leq\sum_{i=1}^nH(X_i)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 161 / 2 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-161/2/i/solution