# 6.6: Chapter Key Items

Normal Distribution
a continuous random variable $$(RV)$$ with pdf $$f(x) =$$

$\frac{1}{\sigma \sqrt{2 \pi}} \mathrm{e}^{\frac{-(x-\mu)^{2}}{2 \sigma^{2}}}\nonumber$

, where $$\mu$$ is the mean of the distribution and $$\sigma$$ is the standard deviation; notation: $$X \sim N(\mu, \sigma)$$. If $$\mu = 0$$ and $$\sigma = 1$$, the $$RV$$, $$Z$$, is called the standard normal distribution.
Standard Normal Distribution
a continuous random variable $$(RV) X \sim N(0, 1)$$; when $$X$$ follows the standard normal distribution, it is often noted as $$Z \sim N(0, 1)$$.
z-score
the linear transformation of the form $$z=\frac{x-\mu}{\sigma}$$ or written as $$z=\frac{|x-\mu|}{\sigma}$$; if this transformation is applied to any normal distribution $$X \sim N(\mu, \sigma)$$ the result is the standard normal distribution $$Z \sim N(0,1)$$. If this transformation is applied to any specific value $$x$$ of the $$RV$$ with mean $$\mu$$ and standard deviation $$\sigma$$, the result is called the z-score of $$x$$. The z-score allows us to compare data that are normally distributed but scaled differently. A z-score is the number of standard deviations a particular $$x$$ is away from its mean value.