1 Introduction to Stats: Expected values

The empirical average is a very intuitive idea; it’s the middle of our data in a sense. But, what is it estimating? We can formally define the middle of a population distribution. This is the expected value. Expected values are very useful for characterizing populations and usually represent the first thing that we’re interested in estimating.

Statistical inference: The process of making conclusions about populations from noisy data that was drawn from it.

Sample expected values (Sample mean and Variance) will estimate the population versions. The expected value or mean of a random variable is the of it’s distribution.

For a discrete random variable \(x\) with a probability mass function \(p(x)\) is simply the sum of all of the values that x can take, multiplied by the probability that it will take these values:

\[E[X] = \sum_x xp(x)\]

\(E[X]\) represents the of mass of a collection of locations and weights, \({x,p(x)}\).
The of mass of the data is the empirical mean \(\bar{X}\)

\[\bar{X} = \sum_{i=1}^n x_ip(x_i)\]