expected value

$\mathbb{E}$ stands for the expectation, which is a mathematical operator that calculates the average value of a random variable over many possible outcomes.

In short, it’s the mean of a probability distribution:

$$\text{displaylines}\begin{array}{r}\mu =\mathbb{E}\left[X\right]=\sum _{i=1}^{n}P(X=x_i)\cdot x_i\end{array}$$

$X\dots$ random variable (distribution)
$x_i\dots$ possible outcomes of the random variable $X$
The expected value of a random variable is like saying you go through all of the possible outcomes and multiply the probability of that outcome times the value of that variable.

For continous random variables (described by PDFs), the expected value is:

$$\text{displaylines}\begin{array}{r}\mathbb{E}[X]={\int }_{-\infty }^{\infty }x\cdot f(x)dx\end{array}$$

The integral exetends over all possible values of $X$.
In a more general notation:

$$\mathbb{E}[g(X)]={\int }_{x}f(x)g(x)$$

where $f(x)$ is the probability density function and $g(x)$ is the expetation function of the random variable and ${\int }_x$ is shorthand for integrating over all possible values of $X$.

Expectations can be approximated by with a sample mean:

$$\mathbb{E}[X]\approx \frac{1}{N}\sum _{i=1}^N{x}_i$$

If we draw from a particular distribution $P$, we write it like this

$$\underset{x\sim P}{\mathbb{E}}[X]={\int }_{x}P(x)x$$

Properties

The operation is linear:

$$\text{displaylines}\begin{array}{r}\mathbb{E}[X+Y]=\mathbb{E}[X]+\mathbb{E}[Y]\end{array}$$

The expected value is a constant and constants stay constants:

$$\text{displaylines}\begin{array}{r}\mathbb{E}[\mathbb{E}[X]]=\mathbb{E}[X]\end{array}$$

If the distribution is uniform, the expected value simplifies to the arithmetic mean of the values.

References

https://chat.openai.com/c/92cd3571-92b1-4c8f-8345-b62a753888e3

probability