Question

example 4.8
compute the variance and standard deviation of the discrete probability distribution and interpret in context of netflix movies streamed each month.
\

$$\begin{tabular}{cc} x&p(x)\\\\\\hline0&0.06\\\\1&0.58\\\\2&0.22\\\\3&0.10\\\\4&0.03\\\\5&0.01\\\\\\hline\\end{tabular}$$

Explanation:

Step1: Calculate the mean $\mu$

$\mu=\sum_{x}x\cdot P(x)=0\times0.06 + 1\times0.58+2\times0.22 + 3\times0.10+4\times0.03+5\times0.01$
$=0 + 0.58+0.44 + 0.30+0.12+0.05=1.49$

Step2: Calculate the variance $\sigma^{2}$

$\sigma^{2}=\sum_{x}(x - \mu)^{2}\cdot P(x)$
$=(0 - 1.49)^{2}\times0.06+(1 - 1.49)^{2}\times0.58+(2 - 1.49)^{2}\times0.22+(3 - 1.49)^{2}\times0.10+(4 - 1.49)^{2}\times0.03+(5 - 1.49)^{2}\times0.01$
$=(- 1.49)^{2}\times0.06+(-0.49)^{2}\times0.58+(0.51)^{2}\times0.22+(1.51)^{2}\times0.10+(2.51)^{2}\times0.03+(3.51)^{2}\times0.01$
$=2.2201\times0.06 + 0.2401\times0.58+0.2601\times0.22+2.2801\times0.10+6.3001\times0.03+12.3201\times0.01$
$=0.133206+0.139258+0.057222+0.22801+0.189003+0.123201$
$=0.869899\approx0.87$

Step3: Calculate the standard - deviation $\sigma$

$\sigma=\sqrt{\sigma^{2}}=\sqrt{0.869899}\approx0.93$

Answer:

The variance is approximately $0.87$ and the standard deviation is approximately $0.93$. This means that, on average, the number of Netflix movies streamed each month deviates from the mean of $1.49$ by about $0.93$ movies, and the spread of the distribution of the number of movies streamed is measured by the variance of $0.87$.