Can Positive Values of Observations imply Population Positive Mean? Suppose that we have N-observations for a random variable $V$ from a population:
$$\{v_1,\;\ldots,\;v_N\}$$
, and all of the observations is positive: $v_i>0\quad \forall i=1,\;\ldots,\;N$.
Here, assume that the observations are "randomly sampled" (i.i.d.)
I think, from the positive values of observations, it is reasonable to guess that the population mean $\mathbb{E}[V]$ is also positive because the observations may represent the population distribution due to the random sampling assumption.
But, I am also skeptical about whether a statement for data can imply a statement for a population.
So, is it possible to derive $\mathbb{E}[V]>0$ from $v_i>0\quad \forall i=1,\;\ldots,N$?
In other words, can we mathematically prove the following implication?
$$v_i>0\quad \forall i=1,\;\ldots,N \quad \Longrightarrow \quad \mathbb{E}[V]>0$$
(Note that I am not asking an inference question. I want to know whether we can prove that under the random sampling assumption.)

No. A sample generally does not allow you to "prove" anything about the underlying population with 100% certainty, it just allows you to infer it with varying levels of confidence. Suppose you have a distribution that truly has a negative expected value, but has non-zero probability for positive values. No matter how many samples you pick, it's possible that every single observed value is positive, despite the fact that the expected value is actually negative. It becomes increasingly unlikely to that the expected value is negative as you observe more and more positive samples, but you can never be 100% sure.
Suppose you have a coin with a -2 on one face, and a +1 on the other. The true expected value of the coin flip is -0.5, and yet it's possible to conduct a series of coin flips and observe the +1 face every single time. Observing only +1's does not guarantee the expected value is positive, no matter how many you observe.
