Chapter 1 Continuous Distributions

1.1 Standard Normal Density Function

The standard normal density function is \[f(x) = \dfrac{1}{\sqrt{2\pi}}e^{{-x^2/2}}, \ \ \ -\infty < x < +\infty\]

Example

library(metricsgraphics)
xs = seq(from = -3, to = 3, by = 0.01)
df <- data.frame(
x = xs,
y = dnorm(xs))

df %>%
mjs_plot(x = x, 
         y = y, 
         title = "Standard Normal Density Function", 
         description = "This is a graph of the standard normal density function") %>%
mjs_line()

1.2 The standard Normal Distribution Function

The standard normal distribution function is defined as

\[ \begin{eqnarray*} F(x) & = & P[X \leq x]\\ & & \\ & = & \int\limits_{-\infty}^x f(x) \ dx \\ & & \\ & = & \dfrac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^x e^{-t^2/2} \ dt \\ \end{eqnarray*} \]

library(metricsgraphics)
xs = seq(from = -3, to = 3, by = 0.01)
df <- data.frame(
x = xs,
y = pnorm(xs))

df %>%
mjs_plot(x = x, 
         y = y, 
         title = "Standard Normal Distribution Function", 
         description = "This is a graph of the standard normal distribution function") %>%
mjs_line()