Let $N_1,N_2$ be two random variables taking positive integer values ($\geq 1$), and it's always true that $N_1\leq N_2$. Let $k\geq1$ be a constant integer. Can someone confirm with me whether it's true that the conditional expectations $E(N_1 - k | N_1\geq k)\leq E(N_2 - k | N_2\geq k)$?
The reason why I think it's true is that: $E(N_1 - k | N_1\geq k)\leq E(N_2 - k | N_1\geq k)\leq E(N_2 - k | N_2\geq k)$. I am not sure whether the second inequality is correct.
Thanks for your help!
This is actually not correct. Here is a counter example.
Let $\mathbb{P}(N_2 = 1) = \mathbb{P}(N_2 = 0) = 1/2$ and $\mathbb{P}(N_1 = 1) = \mathbb{P}(N_1 = -1) = 1/2$. Take their joint distribution to be $\mathbb{P}(N_2 = 1, N_1 = 1) = \mathbb{P}(N_2=0, N_1=-1) = 1/2$. Then $N_2 \geq N_1$ always.
Now set $k=0$. We can calculate \begin{align} \mathbb{E}[N_2 - k|N_2 \geq k] &= \mathbb{E}[N_2|N_2 \geq 0] = 1/2 \end{align} \begin{align} \mathbb{E}[N_1 - k|N_1 \geq k] &= \mathbb{E}[N_1|N_1 \geq 0] = 1 \end{align}
and therefore $\mathbb{E}[N_2 - k|N_2 \geq k] < \mathbb{E}[N_1 - k|N_1 \geq k]$.
The idea of this example is that you want to construct a random variable $N_1$ such that $N_1$ is usually much less than $k$, but when it exceeds $k$ it does so by a lot. This forces the conditional expectation of $N_1 - k$ to be large on the event $N_1 \geq k$. Now compare this to $N_2$ where $N_2 \geq k$ always. What happens here is that there are still some cases where $N_2 - k$ is large, but in the expectation they are diluted by the cases where $N_2 - k$ is small. This dilution is the source of the issue.