Addition of distributions

Question

Give an example of discrete random variables ξ, η, ζ such that $F_{ξ}$ ≡ $F_{η}$, but $F_{ξ+ ζ}$ $\ne$ $F_{η + ζ}$.

Answer 1

I'll give you a start and elaborate further if need be.

Consider the space $\Omega=\{ 1,2 \}$ with the probability:

$$ \mathbb{P}(\{1\}), \mathbb{P}(\{2\})=\frac{1}{2} $$

Define $\xi:\Omega\rightarrow \mathbb{R}$ as: $$ \xi(x)= \begin{cases} 0 & ;x=1 \\ 1 &; x=2 \end{cases} $$

And $\eta :\Omega\rightarrow \mathbb{R}$ as:

$$ \eta(x)= \begin{cases} 1 & ;x=1 \\ 0 &; x=2 \end{cases} $$

Verify that they will satisfy what you need from $\eta$ and $\xi$, and see what happens for $\zeta:\Omega \rightarrow \mathbb{R}$, where:

$$ \zeta(x)= \begin{cases} -1 & ;x=1 \\ 0 &; x=2 \end{cases} $$

Then:

$$ F_{\zeta+\eta}(x)=\begin{cases}0 &; x<0 \\ 1& ;x\geq 0 \end{cases} \quad \text{and} \quad F_{\zeta+\xi}(x)= \begin{cases} 0 & ;x<-1 \\ \frac{1}{2} & ; -1\leq x<1 \\ 1 & ;x\geq 1 \end{cases} $$