Before Sampling vs After Sampling

There is a famous sentence in statistics.

Before sampling, X is a random variable.

After sampling, X is a number.

These are not useless sentence.

Before sample, $X, E(X), Var(X), Cov(X,Y), r(X,Y)$ are random variables or random vectors.

After sample, $X$ is a number or a vector and $E(X), Var(X), Cov(X,Y), r(X,Y)$ are numbers.

An Example of After Sampling

	X	Y
trial 1	1	2
trial 2	2	4
trial 3	4	8
trial 4	3	6
trial 5	2	4

We can easily get

$$
E(X) = \frac{12}{5} = 2.4
$$

$$
E(Y) = \frac{24}{5} = 4.8
$$

$$
Cov(X,Y) = \frac{242}{125} = 1.936
$$

An Example of Before Sampling

So, based on the data, the simplest model (completely over-fitting) is

Probability	X
$\frac{1}{5}$	1
$\frac{2}{5}$	2
$\frac{1}{5}$	3
$\frac{1}{5}$	4

Probability	Y
$\frac{1}{5}$	2
$\frac{2}{5}$	4
$\frac{1}{5}$	6
$\frac{1}{5}$	8

Probability	(X,Y)
$\frac{1}{5}$	(1,2)
$\frac{2}{5}$	(2,4)
$\frac{1}{5}$	(3,6)
$\frac{1}{5}$	(4,8)

We can easily get

$$
E(X) = E_{w}(X(w)) = \frac{12}{5} = 2.4
$$

$$
E(Y) = E_{w}(Y(w)) = \frac{24}{5} = 4.8
$$

$$
Cov(X,Y) = Cov_{w}(X(w),Y(w)) = E_{w}((X-E(X))(Y-E(Y))) =
E(XY) - E(X)E(Y) = \frac{242}{125} = 1.936
$$

THINK

w vs t

For stochastic process $X(w,t)$ and $Y(w,t)$,

We can calculate $F(t) = Cov_{w}(X(w,t),Y(w,t))$

We can also calculate $F(w) = Cov_{t}(X(w,t),Y(w,t))$

Connection with Calculus

Random variable is the same as independent variable or dependent variable.

Number is a variable which only has 1 value.