# Developing a Probability Distribution from Empirical Data

In real-world situations, statisticians obtain data by means of observation and experimental methods. Data obtained in this manner is called empirical data.

A
probability distribution
obtained by means of observation and experimental methods is referred to as an
**
empirical probability distribution
**
, or a relative
frequency distribution
based on observation.

**
Example:
**

Let $X$ be the number of movies a high school student watches in a given month.

A survey conducted at one particular high school in the month of December gives by the following table:

Number of Movies Watched | $1$ | $2$ | $3$ | $4$ | $5$ |

$\%$ of students | $17$ | $28$ | $34$ | $15$ | $6$ |

If we assume that the students at this high school are typical, and that December is a typical month, then

The probability that a high school student will watch $1$ movie per month is $17\%$ ;

The probability that a high school student will watch $2$ movies per month is $28\%$ ;

etc.

Use this empirical probability distribution to find the expected value for the number of movies a high school student will watch in a month.

Convert the percentages to decimals.

$17\%=0.17$

Use the weighted average formula.

$E\left(x\right)={\displaystyle \sum {x}_{i}\cdot P\left({x}_{i}\right)}$

$\begin{array}{l}=\left(1\right)\cdot \left(0.17\right)+\left(2\right)\cdot \left(0.28\right)+\left(3\right)\cdot \left(0.34\right)+\left(4\right)\cdot \left(0.15\right)+\left(5\right)\cdot \left(0.06\right)\\ =0.17+0.56+1.02+0.90+0.30\\ =2.95\end{array}$

So, we can expect the average high school student to watch $2.95$ movies per month.