We define mixture distribution as follows (see wiki):
Given a finite set of probability density functions $p_1(x), \ldots, p_n(x)$, or corresponding cumulative distribution functions $P_1(x), \ldots, P_n(x)$ and weights $w_1, \ldots, w_n$ such that $w_i \ge 0$ and $\sum w_i = 1$, the mixture distribution can be represented by writing either the density, $f$, or the distribution function, $F$, as a sum (which in both cases is a convex combination):
$\displaystyle F(x) = \sum_{i=1}^n \, w_i \, P_i(x),$
$\displaystyle f(x) = \sum_{i=1}^n \, w_i \, p_i(x).$
Next, mixed distribution is defined as follows (see here or here):
Let $X$ be a random variable taking values in $S \subseteq \mathbb{R}$. Then $X$ has a distribution of mixed type if $S$ can be partitioned into subsets $D$ and $C$ with the following properties:
a) $D$ is countable and $0 \lt P(X \in D) \lt 1$.
b) $P(X = x) = 0$ for all $x \in C$.
Question.
Is it true that mixture distribution with at least one discrete term ($P_i(x)$) and at least one continuous term ($P_j(x)$) in its CDF is equivalent to mixed distribution? I.e. mixed distribution is just a special case of a mixture distribution?
(I think yes, particularly for $n=2$ we have $w_1 = P(X \in D), w_2 = 1 - w_1$ if $P_1$(x) is discrete and $P_2(x)$ is continuous. But I'm not completely sure...)