Student's t-Distribution
A family of distributions bridging the Gaussian and Cauchy, with tunable tail thickness.
Student's t-distribution bridges the gap between the Gaussian and the Cauchy distribution. It provides a family of distributions with tunable tail thickness, making it invaluable for understanding the spectrum from thin to fat tails.
Polynomial vs Exponential Decay
To understand the t-distribution, we first need to contrast two types of decay: exponential (used by the Gaussian) and polynomial(used by the t-distribution).
Exponential Decay (Gaussian)
The Gaussian distribution decays as:
This decays faster than any polynomial — extremely thin tails.
Polynomial Decay (t-distribution)
The t-distribution decays as:
Read: “one plus x squared over nu, all raised to the minus nu plus one over two”
A polynomial decay — much slower than exponential, creating heavier tails
The parameter ν (nu, “degrees of freedom”) controls how fast the polynomial decays. Lower ν means slower decay and fatter tails.
Explore: Polynomial vs Exponential
Compare polynomial and exponential decay. Notice how polynomial decay stays higher in the tails — this is what makes t-distributions “fat-tailed.”
Polynomial vs Exponential Decay
The t-distribution uses polynomial decay: (1 + x²/ν)−(ν+1)/2. Compare this to the Gaussian's exponential decay: e−x²/2. Polynomial decay is always slower, creating heavier tails.
Polynomial (ν=3): 2.49e-2
Gaussian: 3.35e-4
The ν Parameter as a “Tail Dial”
ν = 1 (Cauchy)
Extremely fat tails. No mean exists. Decays as x−2.
ν = 3 to 5
Moderate fat tails. Similar to financial returns. Mean exists, variance may not.
ν → ∞
Approaches Gaussian. At ν = 30, nearly indistinguishable from normal.
Why polynomial decay matters:
- e−x² drops faster than any polynomial as x → ∞
- This means t-distribution tails are always heavier than Gaussian
- The parameter ν controls how much heavier
- Low ν = slow polynomial decay = extreme events more likely
Why This Creates Fat Tails
For large x, exponential decay (e−x²) drops to zero faster than any polynomial (x−n) ever could. This means:
- Gaussian: extreme events are vanishingly rare
- t-distribution: extreme events remain meaningfully probable
- The gap between them grows as you move further into the tails
The t-Distribution PDF
Student's t-Distribution
A random variable follows a t-distributionwith ν degrees of freedom () if its PDF is:
We write or .
Historical Origin
The t-distribution was introduced by William Sealy Gosset in 1908, writing under the pseudonym “Student” (he worked at Guinness brewery, which forbade publishing). He needed it for quality control with small samples.
In statistics, the t-distribution arises when estimating the mean of a normally distributed population with unknown variance from a small sample. If are i.i.d. Gaussian, then:
where is the sample standard deviation.
Key Properties
Symmetry
The t-distribution is symmetric around 0 (when centered). The mean, median, and mode are all 0 (when the mean exists).
Moment Existence
Like the Pareto, moments exist only under conditions on ν:
- Mean: exists if
- Variance: exists if
- Skewness: = 0 (by symmetry) if
- Kurtosis: exists if
Note that excess kurtosis is , always positive for. The t-distribution always has heavier tails than the Gaussian (except in the limit).
Tail Decay
For large |x|, the t-distribution decays as a power law:
Read: “The probability that absolute X exceeds x decays like x to the minus nu”
The tails follow a power law with exponent equal to the degrees of freedom
Special Cases
ν = 1: Cauchy Distribution
No moments exist — not even the mean! The sample average of Cauchy random variables has the same distribution as a single observation.
ν = 2: Extremely Fat Tails
Mean exists (= 0) but variance is infinite. The distribution is heavier than most financial return data.
ν = 3 to 5: Typical Financial Data
Stock returns often exhibit behavior consistent with . Variance exists but kurtosis may be infinite.
ν → ∞: Gaussian Limit
As ν increases, the t-distribution approaches the Gaussian. At, it's nearly indistinguishable.
The ν Parameter as a Dial
Think of ν as a dial that controls tail thickness:
- Turn it low (ν = 1-2): Extreme fat tails, missing moments
- Middle range (ν = 3-10): Moderate fat tails, some moments exist
- Turn it high (ν → ∞): Thin tails, approaches Gaussian
Explore: t-Distribution Tail Behavior
Adjust the degrees of freedom ν and observe how the distribution changes from extreme fat tails (Cauchy at ν=1) to thin tails (Gaussian as ν→∞).
Fat Tails (ν = 3.0): Variance = 3.00 (finite). Tails decay as x^(-3.0), much slower than Gaussian.
Probability Density Function
Survival Function (Right Tail) — Log-Log Scale
On log-log scale: t-distribution shows power law (straight line, slope = -ν), while Gaussian drops off exponentially (curved, steeper).
Moment Existence (ν = 3.0)
The Cauchy Distribution: A Cautionary Tale
The Cauchy distribution (t with ν = 1) is a famous pathological case:
Sample Mean Instability
Take n i.i.d. Cauchy random variables and compute their average. The average itself follows a Cauchy distribution! Adding more data doesn't help — the sample mean never “settles down.”
This dramatically violates our intuition from the Law of Large Numbers, which requires finite mean.
The Cauchy arises naturally in physics (resonance phenomena) and appears when you take the ratio of two standard normals.
Comparison to Pareto
Both t-distributions and Pareto distributions have power law tails:
| Property | Student's t | Pareto |
|---|---|---|
| Support | All real numbers | |
| Symmetry | Symmetric around 0 | One-sided (only positive tail) |
| Tail decay | ||
| Mean exists if | ||
| Variance exists if |
The t-distribution is useful when you need symmetric fat tails (e.g., modeling returns which can be positive or negative). Pareto is for one-sided phenomena (wealth, city sizes, etc.).
Key Takeaways
- The t-distribution has PDF
- The parameter ν (degrees of freedom) controls tail thickness
- ν = 1 is Cauchy (no mean); ν → ∞ is Gaussian
- Tails decay as power law:
- n-th moment exists only if ν > n
- Provides a continuous spectrum from extreme fat tails to thin tails