1.
Exploratory Data Analysis
1.3. EDA Techniques 1.3.6. Probability Distributions 1.3.6.6. Gallery of Distributions


Probability Density Function 
The general formula for the probability
density function of the normal distribution is
\( f(x) = \frac{e^{(x  \mu)^{2}/(2\sigma^{2}) }} {\sigma\sqrt{2\pi}} \) where μ is the location parameter and σ is the scale parameter. The case where μ = 0 and σ = 1 is called the standard normal distribution. The equation for the standard normal distribution is \( f(x) = \frac{e^{x^{2}/2}} {\sqrt{2\pi}} \) Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function. The following is the plot of the standard normal probability density function.


Cumulative Distribution Function 
The formula for the cumulative distribution function of the standard
normal distribution is
\( F(x) = \int_{\infty}^{x} \frac{e^{x^{2}/2}} {\sqrt{2\pi}} \) Note that this integral does not exist in a simple closed formula. It is computed numerically. The following is the plot of the normal cumulative distribution function.


Percent Point Function 
The formula for the percent point
function of the normal distribution does not exist in
a simple closed formula. It is computed numerically.
The following is the plot of the normal percent point function.


Hazard Function 
The formula for the hazard
function of the normal distribution is
\( h(x) = \frac{\phi(x)} {\Phi(x)} \) where \(\Phi\) is the cumulative distribution function of the standard normal distribution and \(\phi\) is the probability density function of the standard normal distribution. The following is the plot of the normal hazard function.


Cumulative Hazard Function 
The normal cumulative hazard function
can be computed from the normal cumulative distribution function.
The following is the plot of the normal cumulative hazard function.


Survival Function 
The normal survival function
can be computed from the normal cumulative distribution function.
The following is the plot of the normal survival function.


Inverse Survival Function 
The normal inverse survival
function can be computed from the normal percent point function.
The following is the plot of the normal inverse survival function.


Common Statistics 


Parameter Estimation  The location and scale parameters of the normal distribution can be estimated with the sample mean and sample standard deviation, respectively.  
Comments 
For both theoretical and practical reasons, the normal distribution is
probably the most important distribution in statistics. For example,


Theroretical Justification  Central Limit Theorem 
The normal distribution is widely used. Part of the appeal is
that it is well behaved and mathematically tractable. However,
the central limit theorem provides a theoretical basis for why it
has wide applicability.
The central limit theorem basically states that as the sample size (N) becomes large, the following occur:


Software  Most general purpose statistical software programs support at least some of the probability functions for the normal distribution. 