Interpretation of expected value

Question

Let $\mathbb{E}(X)$ be the expected value of a discrete random variable $X$. How do I interpret $\mathbb{E}(X)$?

One of the way to interpret $\mathbb{E}(X)$ is to consider a large number of trails of the experiment, and then take the arithmetic mean of the values taken by $X$. Equivalently, we can think of $\mathbb{E}(X)$ as the center of mass of the distribution of $X$. Are there any other better ways to interpret $\mathbb{E}(X)$ ?

For instance, if we have $X\sim\text{Geom}(p)$, interpreted as the number of Bernoulli trials needed to get one success, then we have $\mathbb{E}(X)=1/p$. If $p=1/10$, then we have $\mathbb{E}(X)=10$. This would mean that, on average it would take $10$ trials to get the first success. I am having a hard time to digest this!

Answer 1

The ways you suggested are certainly correct. Another way to interpret $E(X)$ is as the value you would most often see if you were to observe $X$ over a long period of time.

Answer 2

Consider the sales (in units) of a product over 10 days: \begin{array}{c|l|c} \text{Sales} & \text{# of days, $f$} & \text{$rf$, $p$} \\ \hline 1 & \ \ \ \ \ \ \ \ \ 2 & 0.2 \\ 2 & \ \ \ \ \ \ \ \ \ 5 & 0.5 \\ 3 & \ \ \ \ \ \ \ \ \ 3 & 0.3 \\ \hline & \ \ \ \ \ \ \ \ 10 & 1 \end{array} The arithmetic mean and expected value of sales are equivalent:

$$\bar x=\frac{\sum_{i=1}^3 f_ix_i}{\sum_{i=1}^3 f_i}= \frac{f_1x_1+f_2x_2+f_3x_3}{\sum_{i=1}^3 f_i}=\frac{f_1}{\sum_{i=1}^3 f_i}x_1+\frac{f_2}{\sum_{i=1}^3 f_i}x_2+\frac{f_3}{\sum_{i=1}^3 f_i}x_3=$$ $$p_1x_1+p_2x_2+p_3x_3=\sum_{i=1}^3 p_ix_i=0.2\cdot1+0.5\cdot2+0.3\cdot3=2.1.$$

So, the sales on average are 2.1. Or it is expected to sell 2.1 products.