Lotus BlogTheory - [T14]
The expected value of a sample mean is a tool we can use to correlate an empirical distribution to a theoretical distribution. This is an important process to perform in Statistical Inference, and it involves the study of the probability that a Random Variable has to represent a certain value. For a more intuitive point of view, we can see the Expected Value ($E(X)$) of a random variable as a generalization of the weighted mean. Let’s in fact start from what we already know very well:
$$ Arithmetic Mean = \sum_{i=1}^n{\frac{x_i}{n}} $$
As we know every $x_i$, can be the same to any other $x_i$, but if we wanted to represent this mean only through distinct values, but taking into account repetitions, we could rewrite in the following way:
$$ \sum_{i=1}^n{\frac{x_i}{n}} = \sum_{i=1}^{k}{\frac{x_i \cdot n_i}{n}} $$
It is easy to observe that this looks very much like a weighted mean, but most importantly we introduced the relative frequency in our formula. $\frac{n_i}{n} = f_i$ This highlights the correlation we were looking for, and allows us to define our theoretical $E(X)$ as:
$$ E(X) = \sum_{i=1}^{k}{X_i \cdot P_i} $$