Normal Distribution

Normal Distribution

ideal model for continuous data
If a continuous random variable $X$ follows a normal distribution with mean $\mu$ and standard deviation $\sigma$, this is generally written $X\sim N(\mu ,{\sigma }^2)$
$\mu$ tells you where the center of the curve is, and $\sigma$ gives you the spread.
as ${\sigma }^2$ gets larger, the flatter and wider the normal curve becomes

No matter how far you go out on the graph, the probability density never equals 0.

How to calculate a normal probability

1. determine distribution

2. standardize to N(0,1)

3. look up probabilities

use a z-score table

• given as $P(Z<z)$

X + Y Distribution

If $X\sim N({\mu }_x,{\sigma }_x^2)$ and $Y\sim N({\mu }_y,{\sigma }_y^2)$ , then $X+Y\sim N(\mu ,{\sigma }^2)$ where

• $\mu ={\mu }_x+{\mu }_y$
• ${\sigma }^2={\sigma }_x^2+{\sigma }_y^2$
  If $X\sim N({\mu }_x,{\sigma }_x^2)$ and $Y\sim N({\mu }_y,{\sigma }_y^2)$ , then $X-Y\sim N(\mu ,{\sigma }^2)$ where
• $\mu ={\mu }_x-{\mu }_y$
• ${\sigma }^2={\sigma }_x^2+{\sigma }_y^2$
  Standard score:

Linear Transforms

If $X\sim N(\mu ,{\sigma }^2)$ and $a$ and $b$ are numbers, then

If $X_1,X_2,...,X_n$ are independent observations of $X$ where $X\sim N(\mu ,{\sigma }^2)$ , then

Sources: 1