Question

Explain the relationship between the Frequency distribution table and the discrete random variable and how to use the Frequency distribution table to find the probability density function, and then explain how to calculate the Expectation, the standard deviation, and Coefficient of variation for it, using students’ bicycles in statistics as a practical example

Answer 1

Frequency distribution table shows how many times $n_i$ would you expect appearing of some value $x_i$ of discrete random variable $X$ through the $N$ experiments. If you divide frequency of $x_i$ by total number of measurements $N$, you will obtain normed frequency distribution that is an approximation to probability density function (PDF). When $N \rightarrow \infty$ normed frequency distribution becomes PDF. Frequency distribution table has two arrays: $x_i ; n_i$ .

To find expectation (or mean) you need to calculate $\bar{x}=\frac{\sum x_i n_i}{N} = \mu$ .

Standard deviation: $\sigma =\sqrt{\frac{\sum n_i(x_i-\bar{x})^2}{N}}$.

Coefficient of variation: $c_V = \frac{\sigma}{\mu}$.