Recursively defined probability, finding expected value $\Bbb E[X]$.

Question

Consider a pmf where $P(X=i)$ is defined for $i = 0, 1, 2$ and for some constant $c$, $$cP(X=i-1)=P(X=i)$$ for $i = 1, 2$. What is the expected value $\Bbb E[X]$ of this pmf?

Answer 1

If you substitute $i=1,2$ in the given relation you get the system \begin{cases}cP(X=1-1)=P(X=1)\\cP(X=2-1)=P(X=2)\\P(X=0)+P(X=1)+P(X=2)=1\end{cases} where the last equation stands because all probabilities must sum up to $1$. This system of $3$ equations in $3$ unknowns gives $$\begin{cases}P(X=1)=cP(X=0)\\P(X=2)=c^2P(X=0)\\P(X=0)(1+c+c^2)=1\end{cases}\implies\begin{cases}P(X=1)=\frac{c}{1+c+c^2}\\P(X=2)=\frac{c^2}{1+c+c^2}\\P(X=0)=\frac{1}{1+c+c^2}\end{cases}$$ Hence $$E[X]=0\cdot P(X=0)+1\cdot P(X=1)+2\cdot P(X=2)=\frac{c+2c^2}{1+c+c^2}$$