The Normal Distribution

The ubiquitous 'bell curve' that forms the bedrock of modern statistics.

The Bell Curve

The Normal (or Gaussian) distribution is arguably the most important probability distribution in statistics. It's defined by its mean ( $\mu$ ) and standard deviation ( $\sigma$ ), and its symmetric, bell-shaped curve is instantly recognizable.

Many natural phenomena, from heights and weights to measurement errors, tend to follow a normal distribution. In finance, it's the standard (though often flawed) assumption for modeling asset returns.

Interactive Normal Distribution

Adjust the mean and standard deviation to see how they affect the shape of the curve.

Mean (μ): 0.0

Standard Deviation (σ): 1.0

Mean (

\mu

): 0.00

Variance (

\sigma^2

): 1.00

Core Concepts

Probability Density Function (PDF)

The PDF of a continuous distribution gives the relative likelihood of a random variable being near a given value. The area under the curve between two points gives the probability of the variable falling within that range.

f(x | \mu, \sigma^2) = \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2 }

$\mu$ is the mean, which defines the center of the distribution.
$\sigma$ is the standard deviation, which defines the spread or width of the distribution.

Cumulative Distribution Function (CDF)

The CDF gives the probability that the random variable

X

will take a value less than or equal to

x

F(x) = P(X \le x) = \int_{-\infty}^{x} f(t) dt

There is no simple closed-form solution for the integral of the normal PDF, so its CDF is typically calculated using numerical methods or found by looking up Z-scores in a standard normal table.

Key Derivations

Deriving the Moments

The mean and variance are derived by integrating over the PDF. This involves a standard substitution to simplify the integrals.

Deriving the Expected Value (Mean)

Step 1: Set up the Integral for E[X]

The expected value is the integral of $x \cdot f(x)$ over its domain $(-\infty, \infty)$ .

E[X] = \int_{-\infty}^{\infty} x \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2 } dx

Step 2: Apply u-Substitution

Let $z = (x-\mu)/\sigma$ . Then $x = \mu + z\sigma$ and $dx = \sigma dz$ . Substitute these into the integral.

E[X] = \int_{-\infty}^{\infty} (\mu + z\sigma) \frac{1}{\sigma\sqrt{2\pi}} e^{-z^2/2} (\sigma dz)

The $\sigma$ terms cancel out, simplifying the expression:

E[X] = \int_{-\infty}^{\infty} (\mu + z\sigma) \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz

Step 3: Split the Integral

We can split the integral into two parts:

E[X] = \mu \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz + \sigma \int_{-\infty}^{\infty} z \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz

The first integral is the total area under the standard normal PDF, which is equal to 1.
The second integral is the integral of an odd function ( $z \cdot \text{pdf}(z)$ ) over a symmetric interval, which is equal to 0.

Step 4: Final Result

Combining the results gives us the final formula for the mean.

E[X] = \mu \cdot 1 + \sigma \cdot 0

Final Mean Formula

E[X] = \mu

Deriving the Variance

We use the definition $Var(X) = E[(X-\mu)^2]$ .

Step 1: Set up the Integral for Variance

Var(X) = \int_{-\infty}^{\infty} (x-\mu)^2 \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2 } dx

Step 2: Apply u-Substitution

Again, let $z = (x-\mu)/\sigma$ . Then $x-\mu = z\sigma$ and $dx = \sigma dz$ .

Var(X) = \int_{-\infty}^{\infty} (z\sigma)^2 \frac{1}{\sigma\sqrt{2\pi}} e^{-z^2/2} (\sigma dz)

Simplify by combining terms:

Var(X) = \sigma^2 \int_{-\infty}^{\infty} z^2 \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz

Step 3: Solve with Integration by Parts

The integral $\int z \cdot (z e^{-z^2/2}) dz$ can be solved using integration by parts, where $u=z$ and $dv = z e^{-z^2/2} dz$ . It can be shown that this integral equals 1.

\int_{-\infty}^{\infty} z^2 \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz = 1

Step 4: Final Result

Substituting this result back gives us the variance.

Var(X) = \sigma^2 \cdot 1

Final Variance Formula

Var(X) = \sigma^2

Applications

Quantitative Finance: The Black-Scholes Model

The Black-Scholes model, one of the most famous equations in finance for pricing options, fundamentally assumes that stock returns are normally distributed. Specifically, it assumes that the logarithm of the stock price follows a random walk with a constant drift and volatility, which implies a log-normal distribution for prices and a normal distribution for log-returns.