differences between expected values assume a random variable $X$ taking only natural values (possibly also $0$). Assume two distributions $\pi$ and $\pi'$. Assume an arbitrary $0 < u < 1$.
Take these two inequalities:


*

*$\sum_{n \in N} \pi(n) \cdot u^n \ge \sum_{n \in N} \pi'(n) \cdot u^n$ 

*$\sum_{n \in N} \pi(n) \cdot n \ge \sum_{n \in N} \pi'(n) \cdot n$


The questions I'm trying to solve are the following:


*

*Does 1 imply 2?

*Does 2 imply 1?


Thanks in advance for your help.
 A: The least that can be said is the following :
$\sum_{n\in\mathbb{N}} \pi(n)u^n=g_\pi(u)$ is the generating function of $\pi$. Then your 1 is equivalent to $\forall u\in [0,1]$  :
$$g_\pi(u)\geq g_{\pi'}(u)$$
Your 2 is : $$g'_\pi(1)\geq g'_{\pi'}(1)$$
Edit :
$$\frac{g_\pi(u)-1}{u-1}\leq \frac{g_{\pi'}(u)-1}{u-1}$$
Then $u\rightarrow 1$ : $g'_\pi(1)\leq g'_{\pi'}(1)$
A: For a quick example, given any $X$, try  $X' = X+1$. We see that one statement is true while the other is false, so at least one of those directions doesn't work out no matter what. Ok, actually this can work for $X' = X-1$ too if $X \geq 1$. Is the question correct as stated?
Another example worth considering is to let $Y$ be a [fair] coin flip, and set $X' = Y(X+1) + (1-Y)(X-1)$  (if heads, $X' = X+1$, if tails, $X' = X-1$).

We can actually work with a fixed $u$ in certain situations:
Let $X,X'$ be arbitrary. 
Assume (2) holds, so $E[X'] \leq E[X]$.
Now, let $\pi'_k(j)$ be the conditional distribution given $X=k$.
Further assume that $E[X' | X=k] \leq k$ (strong condition, implies (2)).
For (1), we then have by convexity of $f(x) = u^x$ 
$E[u^{X'}] = \sum_k \pi(k) \sum_j \pi'_k(j) u^{j} \geq \sum_k \pi(k) u^{E[X' | X=k]} \geq \sum_k \pi(k) u^k=E[u^X]$
