Let $X_1, X_2, \dots, X_n$ be a sequence of discrete random variables with PDFs $f_1(x), f_2(x), \dots, f_n(x)$ with corresponding support sets $D_i = \text{supp}(f_i)$ for $i = 1, 2, \dots, n$. Pick a sequence of real numbers $p_1, p_2, \dots, p_n$ that satisfies the following two conditions:
i) $p_i \ge 0$ for each $i = 1, 2, \dots, n$
ii) $\sum_{i=1}^n p_i = 1$.
Show that the function $f(x) = \sum_{i=1}^n p_i f_i(x)$ is a PDF.
