Normal Distribution

The normal distribution — also called the Gaussian distribution or the bell curve — is the most important probability distribution in statistics. It describes the distribution of a random variable that clusters around a central value, with deviations from the center becoming progressively less likely in a symmetric pattern.

The distribution is completely described by two parameters:

• μ (mu) — the mean: The center of the distribution, where the peak occurs.
• σ (sigma) — the standard deviation: Controls the spread. A small σ means values cluster tightly around the mean; a large σ means they are more dispersed.

When μ = 0 and σ = 1, it is called the standard normal distribution, denoted Z ~ N(0, 1). Any normal variable can be converted to a standard normal by subtracting the mean and dividing by the standard deviation: Z = (X - μ) / σ.

The normal distribution appears everywhere in nature and science because of the Central Limit Theorem: whenever you average or sum many independent random variables, the result tends toward normality — regardless of the original distribution.

The normal distribution has several properties that make it mathematically convenient and practically important:

• Symmetry: The distribution is perfectly symmetric around the mean. P(X > μ + a) = P(X < μ - a) for any a.
• The 68-95-99.7 rule: Approximately 68% of values fall within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. This is the most useful rule of thumb in statistics.
• Linear combinations: If X ~ N(μ₁, σ₁²) and Y ~ N(μ₂, σ₂²) are independent, then X + Y ~ N(μ₁ + μ₂, σ₁² + σ₂²). Normal distributions are "closed under addition."
• Fully determined by mean and variance: Unlike many distributions, the normal distribution has no skewness (it's symmetric) and its kurtosis is fixed at 3. All information is captured by μ and σ.
• Maximum entropy: Among all distributions with a given mean and variance, the normal distribution has the maximum entropy — it is the "least informative" distribution, making it the natural default when you only know the mean and variance.

The normal distribution is the workhorse distribution of quantitative finance:

• Black-Scholes model: Assumes that log-returns of the stock are normally distributed (equivalently, that stock prices follow a lognormal distribution). The terms N(d₁) and N(d₂) in the Black-Scholes formula are cumulative normal distribution values.
• Value at Risk: Parametric VaR assumes portfolio returns are normally distributed, using the z-score to compute the loss at a given confidence level. The 99% VaR corresponds to z = 2.33.
• Portfolio theory: Modern Portfolio Theory (Markowitz) assumes returns are normally distributed, which means the entire risk-return tradeoff can be captured by mean and variance alone.
• Statistical testing: Z-tests and t-tests for evaluating trading signals rely on normality (justified by the CLT for large samples).

Example: A portfolio has daily returns with mean 0.05% and standard deviation 1.2%. What is the probability of losing more than 3% in one day?

Z = (-3% - 0.05%) / 1.2% = -2.54. P(Z < -2.54) = 0.0055 = 0.55%.

Under the normal assumption, a 3% daily loss happens about once every 180 trading days. In reality, such losses occur more frequently due to fat tails — a critical limitation of the normal assumption.

While the normal distribution is mathematically convenient, real financial returns deviate from normality in important ways:

• Fat tails (excess kurtosis): Extreme returns happen much more often than the normal distribution predicts. A "6-sigma event" under normality should occur about once every 1.5 million days (roughly once in 6,000 years). In financial markets, 6-sigma events happen every few years. The 2008 financial crisis involved returns that were 10+ sigma events under normal assumptions.
• Negative skewness: Stock return distributions are slightly left-skewed — large negative returns are more common than large positive returns of the same magnitude. The normal distribution is symmetric and cannot capture this.
• Volatility clustering: Large returns tend to be followed by large returns (of either sign), violating the independence assumption underlying many normal models.

These deviations explain why the volatility smile exists (the market prices in fatter tails than Black-Scholes assumes), why VaR underestimates tail risk when assuming normality, and why quant firms use more sophisticated models (Student-t distributions, GARCH models, extreme value theory) for risk management.

The practical takeaway: use the normal distribution as a first approximation, but always be aware of its limitations in the tails.