Let $\phi_1,\ldots,\phi_n$ denote characteristic functions for distributions on the real line. Let $a_1,\ldots,a_n$ denote nonnegative constants such that $a_1+\ldots+a_n = 1$. Show that
$$\hspace{20mm} \phi(t) = \sum_{j=1}^{n}a_j\phi_j(t), \hspace{8mm}-\infty<t<\infty$$
Attempt:
Let $a = 1$.
$$\phi(t) = \mathrm{E}\left[\exp(aitX)\right] = \int_{-\infty}^{\infty} \exp(aitX)\times p(x)\,\mathrm{d}x $$
I'm not sure how to proceed. I would assume that the additive property of the characteristic functions has something to do with the exponential components.