expectation is the calculation of the “intended” or “target” value given a random variable:

\begin{equation} \mathbb{E}[M] = \sum_{x} x\ p(X=x) \end{equation}

- Standardize variables to \(z\) by dividing
- The correlation is simply their “product”: means of positive and negative groups

The expectation is the average of the counts of the data you have.

## properties of expectation

these holds **REGARDLESS** of whether or not the variables you are doing is independent, IID, etc.

### Linearity in the first slot

expectation has additivity and homogeneity.

\begin{equation} \mathbb{E}[aX+b] = a\mathbb{E}[X] + b \end{equation}

### Closure under expectation

\begin{equation} E[X+Y] = E[X]+E[Y] \end{equation}

### Unconscious statistician

\begin{equation} \mathbb{E}[g(x)] = \sum_{x \in X}^{} g(x) P(X=x) \end{equation}

whereby, if \(g\) is a normal function, you can just add up all the possible output. This property can be used to show the firts results.

## conditional expectation

We can perform expectation via conditional probability.

\begin{equation} E[X|Y=y] = \sum_{x}^{} x \cdot p(X=x|Y=y) \end{equation}

## law of total expectation

\begin{equation} \mathbb{E}[X] = \sum_{y}^{}\mathbb{E}[X|Y=y] P(Y=y) \end{equation}

what is the “background variable”? the \(y\) value above.