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

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 $$F(z)=0$$ $$\forall z\leq 0$$.

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.

Question: is the claim correct?

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 .
• if this is for applied purposes, the $F_1$ and $F_2$ would likely be truncated yes – phdmba7of12 Feb 7 at 12:00