Normal Distribution

Type: Continuous probability distribution Also called: Gaussian distribution, bell curve Notation: $X\sim N(\mu ,{\sigma }^2)$

Key Properties

• Symmetric around mean $\mu$
• Fully described by two parameters: $\mu$ (mean) and ${\sigma }^2$ (variance)
• Tails extend to $\pm \infty$ but never touch zero
• Mean = Median = Mode

Confidence Intervals

Interval	Coverage
$\mu \pm 1\sigma$	68.27%
$\mu \pm 1.65\sigma$	90%
$\mu \pm 1.96\sigma$	95%
$\mu \pm 2.58\sigma$	99%

Standard Normal ($Z$)

$Z=\frac{X-\mu }{\sigma }\sim N(0,1)$

Related Distributions

Distribution	Relationship
Lognormal	If $X\sim N$, then $e^X$ is lognormal
t-distribution	Approaches normal as $df\to \infty$
Chi-square	Sum of squared standard normals
F-distribution	Ratio of two chi-square/df

Role in CFA Quant

The normal distribution is the foundation of:

• M05 — Portfolio return modeling, Safety-First ratio
• M06 — Lognormal asset pricing, Monte Carlo
• M07 — CLT, confidence intervals
• M08 — z-test, t-test
• M10 — Normality assumption of residuals

Limitation for Finance

• Real asset returns exhibit fat tails (leptokurtosis) and skewness
• Normal distribution underestimates probability of extreme events
• This is why M03 teaches skewness and kurtosis