Series of probabilities

Question

Consider $Y_k=X_1+...+X_k $, where $X_k \in \mathbb{N}_0$ are i.i.d random variables and $E[X_1]<1$

$$\sum_{j=1}^{\infty} P(Y_j=j) \overset{!}{=}1 $$

How can I verify that this equation is true or does this not hold?

Answer 1

Take $X_i$ iid Bernoulli s.t. $$\mathbb P\{X_i=1\}=p\in (0,1)\setminus \left\{\frac{1}{2}\right\}.$$

Then, $\mathbb E[X_1]=p<1$ and $Y_k$ are Binomial $(k,p)$. Therefore, $$\mathbb P\{Y_j=j\}=p^j.$$

Furthermore $$\sum_{j=1}^\infty p^j=\frac{p}{1-p}\neq 1.$$