Shape

PDF

The normal PDF is given by

\[ f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}; -\infty <x<\infty \]

• First discovered by Abraham De Moivre

Binomial to Normal

• Number of trial (n) is large (\(n \rightarrow \infty\))
• \(P(S) \approx P(\bar S\)), where \(S =\) Success

Empirical Rule

68‑95‑99.7 Rule

Properties

• Bell-shaped
• Symmetrical about mean
• \(\beta_1 = 0, \beta_2 = 3\)
• \(\mu = Me = Mo\)
• Area under whole curve is 1
• The curve on either side of mean extends up to infinity.

More Properties

• Empirical rule
• Linear combination of Normal \(\rightarrow\) normal
• \(MD(\bar X) \frac 45\sigma\)
• \(QD = \frac 23\sigma\)
• \(M_X(t) = \mathbb{E}[e^{tX}] = \exp\left(\mu t + \frac{1}{2} \sigma^2 t^2\right) -\infty \le t \le \infty\)