Normal Distribution

Figure 3: Normal curves with $ \mu _1 < \mu _2$ and $ \sigma _1 = \sigma _2$.
\includegraphics[scale=0.3]{figures/06-03}

Figure 4: Normal curves with $ \mu _1 =\mu _2$ and $ \sigma _1 < \sigma _2$.
\includegraphics[scale=1.2]{figures/06-04}

Figure 5: Normal curves with $ \mu _1 < \mu _2$ and $ \sigma _1 < \sigma _2$.
\includegraphics[scale=1]{figures/06-05}

The properties of the normal curve

  1. The mode, which is the point on the horizontal axis where the curve is a maximum, occurs at $ x =\mu$.
  2. The curve is symmetric about a vertical axis through the mean $ \mu$.
  3. The curve has its points of inflection at $ x =\mu \pm \sigma$ , is concave downward if $ \mu-\sigma<X<\mu+\sigma$, and is concave upward otherwise.
  4. The normal curve approaches the horizontal axis asymptotically as we proceed in either direction away from the mean.
  5. The total area under the curve and above the horizontal axis is equal to 1.
  6. Both tails become dramatically thin beyond $ \pm 3\sigma$ from the mean $ \mu$.

Cem Ozdogan 2010-04-21