By introducing the concepts of weighted pushdown automata and weighted normalized pushdown automata over valuation monoid, the equivalence of weighted pushdown automata and weighted normalized pushdown automata over double unitary valuation monoid is proved.It is shown that the fact that weighted pushdown automata can accept the same formal power series by final states and by empty stack at the same time.The closed properties of weighted context-free languages under some regular operations such as sum, concatenation and positive Kleene closure are dealt with as well. The main conclusions show that whether those properties of weighted pushdown automata are valid or not doesn't rely on the distributivity or associativity of valuation monoid.

valuation monoid; double unital valuation monoid;weighted pushdown automaton; weighted context-free language