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t-Distribution

The t-distribution (Student's t-distribution) arises when estimating the mean of a normally distributed population using a small sample with unknown variance. It has heavier tails than the standard normal distribution, which accounts for the additional uncertainty introduced by estimating the variance from the data. The t-distribution is the basis for t-tests and confidence intervals in classical statistics.

Mental Model

The t-distribution is a "cautious normal." When you estimate variance from a small sample, there is extra uncertainty, so the distribution has heavier tails -- extreme values are more probable than a normal would predict. As the sample size (degrees of freedom) grows, this extra uncertainty vanishes and the t-distribution converges to the standard normal.

Mathematical Definition

If \(Z \sim N(0, 1)\) and \(V \sim \chi^2(\nu)\) are independent, then the random variable

\[ T = \frac{Z}{\sqrt{V/\nu}} \]

follows a t-distribution with \(\nu\) degrees of freedom, written \(T \sim t(\nu)\).

The probability density function is:

\[ f(x) = \frac{\Gamma\!\left(\frac{\nu + 1}{2}\right)}{\sqrt{\nu\pi}\;\Gamma\!\left(\frac{\nu}{2}\right)} \left(1 + \frac{x^2}{\nu}\right)^{-(\nu + 1)/2} \]

where \(\Gamma(\cdot)\) is the gamma function and \(\nu > 0\) is the degrees of freedom parameter.

Usage in scipy.stats

The scipy.stats.t distribution object takes the degrees of freedom df (\(\nu\)):

```python import scipy.stats as stats import numpy as np import matplotlib.pyplot as plt

df = 5 a = stats.t(df)

print(f"Mean: {a.mean():.4f}") # 0 (for ν > 1) print(f"Variance: {a.var():.4f}") # ν/(ν-2) = 5/3

x = np.linspace(-5, 5, 200) y_pdf = a.pdf(x) y_cdf = a.cdf(x)

plt.plot(x, y_pdf, label='PDF') plt.plot(x, y_cdf, label='CDF') plt.legend() plt.title(f't-Distribution (ν={df})') plt.xlabel('x') plt.ylabel('Density / Probability') plt.show() ```

Comparison with the Normal Distribution

The t-distribution is symmetric about zero like the standard normal, but it has heavier tails. As \(\nu\) increases, the t-distribution converges to the standard normal:

```python import scipy.stats as stats import numpy as np import matplotlib.pyplot as plt

x = np.linspace(-5, 5, 200) plt.plot(x, stats.norm.pdf(x), 'k--', linewidth=2, label='N(0,1)') for df in [1, 3, 5, 10, 30]: plt.plot(x, stats.t(df).pdf(x), label=f'ν={df}')

plt.legend() plt.title('t-Distribution PDFs vs Standard Normal') plt.xlabel('x') plt.ylabel('f(x)') plt.show() ```

The case \(\nu = 1\) is the Cauchy distribution, which has such heavy tails that neither the mean nor the variance exists.

Key Properties

  • Mean: \(E[T] = 0\) for \(\nu > 1\) (undefined for \(\nu \le 1\))
  • Variance: \(\text{Var}(T) = \dfrac{\nu}{\nu - 2}\) for \(\nu > 2\) (infinite for \(1 < \nu \le 2\))
  • Symmetry: The distribution is symmetric about 0
  • Heavy tails: For any finite \(\nu\), the tails decay as a power law \(|x|^{-(\nu+1)}\), slower than the exponential decay of the normal
  • Convergence: As \(\nu \to \infty\), \(t(\nu) \to N(0, 1)\)

Special Cases

  • \(\nu = 1\): Cauchy distribution (no finite moments)
  • \(\nu = \infty\): standard normal distribution

Parameters in scipy.stats

Parameter Symbol scipy.stats keyword Default
Degrees of freedom \(\nu\) df (required)
Location — loc 0
Scale — scale 1

Applications in Hypothesis Testing

The t-distribution is used whenever a test statistic involves an estimated standard deviation:

  • One-sample t-test: Testing \(H_0\colon \mu = \mu_0\) using \(T = (\bar{X} - \mu_0)/(S/\sqrt{n})\), which follows \(t(n-1)\) under \(H_0\)
  • Two-sample t-test: Comparing means of two independent groups
  • Confidence intervals: A 95% confidence interval for the mean is \(\bar{X} \pm t_{0.025,\,n-1} \cdot S/\sqrt{n}\)

Critical values are obtained using the PPF:

python alpha = 0.05 df = 20 t_critical = stats.t(df).ppf(1 - alpha / 2) print(f"Two-sided critical value: ±{t_critical:.4f}")

Financial Applications

In finance, the t-distribution is used in modeling asset returns when normal tails are too thin to capture extreme events. The Student-t copula models joint tail dependence between assets. Risk measures such as Value at Risk use the t-distribution for more conservative tail estimates, and GARCH models with t-distributed innovations provide better fits to financial return series.

Summary

The t-distribution arises from the ratio of a standard normal to a scaled chi-square variable and has heavier tails than the normal. In scipy.stats, use stats.t(df) to create a frozen distribution for computing PDFs, CDFs, critical values, and generating random samples.

Exercises

Exercise 1. Plot the PDF of the \(t\)-distribution for \(\nu = 2, 5, 30\) and the standard normal on the same axes over \([-4, 4]\). Observe how the \(t\)-distribution approaches the normal as \(\nu\) increases.

Solution to Exercise 1 
import numpy as np
from scipy import stats
import matplotlib.pyplot as plt

x = np.linspace(-4, 4, 200)
for df in [2, 5, 30]:
    plt.plot(x, stats.t.pdf(x, df), label=f't(df={df})')
plt.plot(x, stats.norm.pdf(x), 'k--', label='Normal')
plt.legend()
plt.title('t-distribution vs Normal')
plt.show()

Exercise 2. For a \(t\)-distribution with 10 degrees of freedom, compute the probability \(P(|T| > 2)\). Compare this with the corresponding probability under the standard normal.

Solution to Exercise 2 
from scipy import stats

p_t = 2 * stats.t.sf(2, df=10)
p_norm = 2 * stats.norm.sf(2)
print(f"P(|T|>2), t(10):  {p_t:.4f}")
print(f"P(|Z|>2), Normal: {p_norm:.4f}")

Exercise 3. Compute the 95th percentile of the \(t\)-distribution for \(\nu = 5, 10, 30, 100\) and the standard normal. Show that the \(t\) percentile converges toward the normal percentile as \(\nu\) grows.

Solution to Exercise 3 
from scipy import stats

for df in [5, 10, 30, 100]:
    print(f"t(df={df:3d}) 95th percentile: {stats.t.ppf(0.95, df):.4f}")
print(f"Normal 95th percentile:     {stats.norm.ppf(0.95):.4f}")