# Sampling Theory and Distributions

## Sampling Theory

Data scientists are required to draw conclusions about a group, a.k.a *population* from a few *samples* of it because
getting the entire population is intractable.
This process of drawing samples is called *sampling*. There are different kinds of sampling , few of which are:

- Random sampling
- Clustered sampling
- Stratified sampling
- Systematic sampling
You can read about them over
here
The drawing of conclusions or
*inference*about the population from the samples is called*statistical inference*.

In this section we will consider two different types of samples:

- Sampling with Replacement
- Sampling without Replacement

## Random Sampling with Replacement

As the name suggests, this is a type of sampling where each member of the population may be included more than once. It’s like picking a ball from an urn and then putting it back into the urn.

## Random Sampling without Replacement

In this type of sampling, each member of the population can be included atmost once. A similar example for this type of sampling would be picking a ball from the urn and not putting it back inside the urn.

## Sampling statistics

A quantity obtained from the sample for the purpose estimating a population parameter is called a *sample statistic* or briefly *statistic*.
Mathematically, a sample statistic for a sample of size $n$ can be defined as a function of the random variables $X_1, X_2,…,X_n$ i.e., $g(X_1, X_2,…,X_n)$.
The function $g(X_1, X_2,…,X_n)$ is another random variable whose values can be represented by $g(x_1, x_2,…,x_n)$.