Normal Distribution & CLT

Easy3–4 hrsNormalCLTZ-scoresLog-Normal

Overview

The Normal distribution is the most important distribution in statistics and finance. The Central Limit Theorem explains why: the average of many independent random variables is approximately normal, regardless of the underlying distribution. The 68-95-99.7 rule is the most useful memorization: 1σ = 68%, 2σ = 95%, 3σ = 99.7%. Stock returns are approximately normal over short periods (though fatter tails in reality).

How to Recognize

→Large sample sizes → invoke CLT for approximate normality
→Sum or average of many independent random variables
→Bell curve, 68-95-99.7 rule, standard deviation questions
→Stock returns, financial P&L, measurement errors

Step-by-Step Approach

1.Standardize: Z = (X − μ)/σ, then use Φ table values
2.CLT: (X̄ − μ)/(σ/√n) → N(0,1) for large n
3.Sum of normals is normal (if independent): μ adds, variance adds
4.Log-normal: X = e^Y where Y~N(μ,σ²). Use for asset prices (always positive)

Key Formulas

\Phi(1)\approx0.841,\ \Phi(1.96)\approx0.975,\ \Phi(2)\approx0.977,\ \Phi(2.58)\approx0.995

0268-95-99.7 rule:

\pm1\sigma,\ \pm2\sigma,\ \pm3\sigma

E[|X|] = \sigma\sqrt{2/\pi} \approx 0.798\sigma

for

X\sim N(0,\sigma^2)

04CLT:

\bar{X} \approx N(\mu, \sigma^2/n)

for large

n

Worked Examples

0Quick Example

Stock N(10%,20%). P(return>30%) = P(Z>(30-10)/20) = P(Z>1) = 1−0.841 = 15.9%.

Problem

A strategy makes independent daily trades with mean profit $100 and standard deviation$ 800. What is the probability of losing money over 100 trading days?

Solution

By CLT: total P&L over 100 days ~ N(100·100, 100·800²) = N(10000, 64000000).

SD of total = 800·√100 = $8,000.

P(total P&L < 0) = P(Z < (0 − 10000)/8000) = P(Z < −1.25).

= 1 − Φ(1.25) ≈ 1 − 0.894 = 10.6%.

Answer

$P(\text{loss over 100 days}) \approx 10.6\%$ . With mean= $10,000 and SD=$ 8,000, there's about 1-in-10 chance of a losing period.

Common Mistakes

!
Confusing σ (population std) with s (sample std). Use s for small samples.
!
Applying CLT with very small n. CLT needs n ≥ 30 as a rough rule for approximately normal distributions.
!
Log-normal: if X is log-normal, E[X] = e^(μ+σ²/2), NOT e^μ. The σ²/2 correction matters.

Practice Problems

Click "Show Answer" to reveal

IQ scores are N(100, 15²). What fraction of people have IQ > 130?

Hint: 130 is how many standard deviations above the mean?

A stock has daily vol of 2% (σ=2%). What is P(stock is down more than 5% in one day)?

Hint: P(X < −5%) = P(Z < −5/2) = P(Z < −2.5).

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