Normal Distribution

A normal or Gaussian distribution is continuous, symmetric, unimodal, bell-shaped, and asymptotic* to the horizontal axis.
The mean, mode, and median of a normal distribution all coincide with the same point in the center of the distribution.
The shorthand notation for a normal distribution with mean #\mu# and standard deviation #\sigma# is #N(\mu, \sigma)#.

Standard Normal Distribution

The Standard Normal Distribution is a normal distribution with mean #\mu=0# and standard deviation #\sigma=1#.

Let #X# be a continuous random variable. If #X\sim N(\mu, \sigma)#, then #Z# follows the Standard Normal Distribution. That is:

\[Z=\cfrac{X-\mu}{\sigma}\sim N(0,1)\]

Binomial Probability Distribution

Let #X# be a binomially distributed random variable with parameters #n# and #p#.
#\phantom{0}#
To calculate the expected value of #X#, make use of the following formula:
\[\mu_{\small{X}} = \mathbb{E}[X]=n\cdot p\]
To calculate the variance of #X#, make use of the following formula:
\[\sigma^2_{\small{X}} = Var[X]=n\cdot p \cdot (1-p)\]
To calculate the standard deviation of #X#, make use of the following formula:
\[\sigma_{\small{X}} = SD[X]=\sqrt{n\cdot p \cdot (1-p)}\]

Sampling Distribution of the Sample Mean

If the conditions for normality are satisfied, the sampling distribution of the sample mean is approximately normally distributed with parameters:

• Expected value of the sample mean: #\mu_{\bar{X}} = \mu#
• Standard error of the sample mean: #\sigma_{\bar{X}} = \cfrac{\sigma}{\sqrt{n}}#

Sampling Distribution of the Sample Proportion

If the conditions for normality are satisfied, the sampling distribution of the sample proportion is approximately normally distributed with parameters:

• Expected value of the sample proportion: #\mu_{\hat{p}} = \pi#
• Standard error of the sample proportion: #\sigma_{\hat{p}} = \sqrt{\cfrac{\pi(1-\pi)}{n}}#