Student t Tails and Moment Boundaries
Compare exact small-degree t tails with a normal law and identify when moments stop existing.
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Student t Tails and Moment Boundaries
The Student t distribution arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown. This module explores its exact tails and moments for small degrees of freedom.
Worked Example
Calculating the exact two-sided tail for 4 degrees of freedom at x = 2.0:
- Identify the degrees of freedom (nu = 4) and probe x (2.0).
- Apply the formula: r = Math.hypot(x, 2), u = x / r, delta = 4 / (r * (r + x)).
- Resulting two-sided tail: delta * delta * (u + 2) / 2 = 0.11611652351681556.
Experiment
We can observe the transition of moment existence and behavior of exact tail values by changing the degrees of freedom in the control panel.
- Set Degrees of freedom to 1. Observe that mean and variance do not exist (Cauchy distribution).
- Change Degrees of freedom to 2. The mean exists (0) but variance is infinite.
- Change Degrees of freedom to 4. Mean is 0, variance is 2. Compare the two-sided tail against the normal distribution tail.
Limitations
This module demonstrates analytic exact forms strictly for degrees of freedom 1, 2, and 4. General numerical approximation for arbitrary degrees of freedom is not implemented here.
References
The following references provide deeper insights into the Student t-distribution, its heavy tails, and moment characteristics.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Student t Tails and Moment Boundaries module.
Try Student t Tails and Moment Boundaries on Riano →