# Let $X$ be a non-negative random variable, and $A_i$ be the event that $X$ is between $[i - 1, i)$.

Let X be a non-negative random variable, and $$A_i$$ be the event that $$i-1 \leq X < i$$.

(a) Show that $$\sum_{i=1}^{\infty}(i-1)I_{Ai} \leq X < \sum_{i=1}^{\infty}iI_{A_i}$$, where $$I_{A_i}$$ = 1 if $$A_i$$ occurs and = 0 otherwise.

Deduce directly that $$\sum_{i=1}^{\infty}P(X \geq i) \leq \mathbf{E}[X] < 1 + \sum_{i=1}^{\infty}P(X \geq i )$$

(b) By considering an exponential distribution for X of rate $$\lambda$$, deduce from part a that for $$\lambda > 0$$, $$\lambda + e^{-\lambda} > 1$$ and $$(\lambda + 1)e^{-\lambda} \leq 1$$.

Finally does the equation proved here have a name? Thank you.

a) If $$j-1 \leq X < j$$, $$\sum_{i=1}^{\infty}(i-1)I_{Ai}= j-1$$ and $$\sum_{i=1}^{\infty}iI_{Ai} = j$$. This is because only the $$i=j$$ term of the sums will contribute. So this proves your inequality. If now sum this inequality for every $$X$$ you get:
$$$$\sum_{i=1}^{\infty}P(X \geq i) \leq \mathbf{E}[X] < 1 + \sum_{i=1}^{\infty}P(X \geq i )$$$$