F-Distribution

Comparing variances between two or more samples.

The "Ratio of Variances" Distribution

The F-distribution is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the Analysis of Variance (ANOVA) and the F-test to compare two variances.

It describes the ratio of two independent chi-squared variables, each divided by their respective degrees of freedom. In practical terms, if you take two samples from normal populations, the ratio of their sample variances follows an F-distribution. This is why it's the cornerstone for checking if the volatility (variance) of two different stocks or trading strategies is significantly different.

Interactive F-Distribution

Adjust the degrees of freedom to see how they influence the shape of the distribution. Notice how it becomes more bell-shaped as the degrees of freedom increase.

Numerator Degrees of Freedom (d₁): 5

Denominator Degrees of Freedom (d₂): 10

Mean (

\mu

): 1.25

Variance (

\sigma^2

): 1.35

Core Concepts

Probability Density Function (PDF)

The PDF is defined by its two parameters: the degrees of freedom of the numerator (

d_1

) and the denominator (

d_2

f(x; d_1, d_2) = \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}}{(d_1 x + d_2)^{d_1 + d_2}}}}{x B(\frac{d_1}{2}, \frac{d_2}{2})}

$d_1 > 0$ is the degrees of freedom for the numerator variance.
$d_2 > 0$ is the degrees of freedom for the denominator variance.
$B$ is the Beta function.

Expected Value & Variance

Expected Value (Mean)

E[X] = \frac{d_2}{d_2 - 2} \quad \text{for } d_2 > 2

The mean is undefined for $d_2 \le 2$ .

Variance

Var(X) = \frac{2d_2^2(d_1+d_2-2)}{d_1(d_2-2)^2(d_2-4)} \quad \text{for } d_2 > 4

The variance is undefined for $d_2 \le 4$ .

Key Derivations

Deriving the Expected Value (Mean)

The mean of an F-distribution is derived from its definition as a ratio of two independent Chi-Squared random variables.

Step 1: Define the F-distribution

An F-distributed random variable $F$ is the ratio of two independent Chi-Squared variables $U_1 \sim \chi^2(d_1)$ and $U_2 \sim \chi^2(d_2)$ , each divided by its degrees of freedom:

F = \frac{U_1 / d_1}{U_2 / d_2}

Step 2: Use the Property of Expectation

Since $U_1$ and $U_2$ are independent, the expectation of their product (or ratio) is the product of their expectations.

E[F] = E\left[\frac{U_1}{d_1}\right] E\left[\frac{d_2}{U_2}\right] = \frac{d_2}{d_1} E[U_1] E\left[\frac{1}{U_2}\right]

Step 3: Substitute Known Expected Values

We know that the mean of a Chi-Squared variable $\chi^2(k)$ is $k$ . Therefore, $E[U_1] = d_1$ .

E[F] = \frac{d_2}{d_1} \cdot d_1 \cdot E\left[\frac{1}{U_2}\right] = d_2 E\left[\frac{1}{U_2}\right]

Step 4: Calculate the Expected Value of 1/U₂

This is the most complex step. It can be shown by integrating $\int_0^\infty \frac{1}{x} f(x; d_2) dx$ (where $f$ is the Chi-Squared PDF) that the expected value of the reciprocal of a Chi-Squared variable is:

E\left[\frac{1}{U_2}\right] = \frac{1}{d_2 - 2}, \quad \text{for } d_2 > 2

This integral only converges if the degrees of freedom are greater than 2, which is why the mean is undefined for $d_2 \le 2$ .

Step 5: Combine Results

Substitute the result from Step 4 back into the equation from Step 3.

E[F] = d_2 \cdot \frac{1}{d_2 - 2}

Final Mean Formula

E[F] = \frac{d_2}{d_2 - 2}