Most Important
It’s the
- most important distribution
- most common distribution
- most widely-used distribution
Shape
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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
Check out here
Poisson to Normal
\(m \rightarrow \infty\)
\[
\lim_{m \to \infty} \text{Pois}(m) \approx \mathcal{N}(m, m)
\] Check out here
Empirical Rule
68‑95‑99.7 Rule
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Empirical Rule Area
- \(P(\mu - \sigma \le \mu + \sigma) = 0.6826\)
- \(P(\mu - 2 \sigma \le \mu + 2 \sigma) = 0.9544\)
- \(P(\mu - 3 \sigma \le \mu + 3 \sigma) = 0.9973\)
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.
- All odd central moments are zero
More Properties
- Empirical rule
- Linear combination of Normal \(\rightarrow\) normal
- \(MD(\bar X) \frac 45\sigma\)
- \(QD = \frac 23\sigma\)
- \(M_X(t) = \mathbb{E}[e^{tX}] = \exp\left(\mu t + \frac{1}{2} \sigma^2 t^2\right) -\infty \le t \le \infty\)
Odd Central Moments
\(\mu_1 = \mu_3 = \cdots = \mu_{2n+1} = 0\)
Mathematically Why? > -
Application or Normal Distribution
- Quality Control in industry
- Most distributions tend to be normal under some assumptions
- Sometimes transformation makes normal
- Used in hypothesis testing (many samples are normally distributed)
Standard Normal
If \(X \sim N(\mu, \sigma ^2), z = \frac{x-\mu}{\sigma} \sim N(0,1)\)
- \(\displaystyle f(z) = \frac{1}{\sqrt{2 \pi}}e^{\frac{-z^2}{2}}\)
- \(E(Z) = 0\)
- \(V(Z) = S(Z) = 0\)
Standard Normal Mean
Let \(X \sim N(\mu, \sigma ^2)\)
- Now let \(\displaystyle Z = \frac{X-\mu}{\sigma}\)
- \(E(Z) = E(\frac{X-Z}{\mu})\)
- = \(\frac{1}{\sigma} E(X-\mu)\)
- = \(\frac{1}{\sigma} [E(X) - E(\mu)]\)
- = \(\frac{1}{\sigma} (\mu - \mu)\)
- 0
Standard Normal Variance
\(V(Z) = V(\frac{X-Z}{\mu})\)
