Expressing a CDF with positive support as a mixture of two components

Question

Consider the two-component mixture $$ F(z)=\lambda F_1(z)+(1-\lambda)F_2(z) $$ where all the $F$'s are CDFs and $\lambda\in [0,1]$.

A1: Assume that $F(z)=0$ $\forall z\leq 0$.

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Claim:

A1 implies that $F(\cdot)$ is not compatible with

• $F_1(\cdot)$ such that $F_1(z)>0$ for some $z\leq 0$ and $\lambda>0$.

• (In light of the previous bullet point) $F_1(\cdot)$ and $F_2(\cdot)$ both Logistic. This is because the Logistic CDF is zero only in the limit for any value of the scale and location.

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Question: is the claim correct?

Answer 1

If for some $z\le 0 $ ,$F(z) = 0$ and $F_{1}(z) > 0$ , then $F_{2}(z) < 0$ , but a CDF can't have this property .

Since the CDF of the logistic distribution has positive values on the negative domain the second claim is also true .