Suppose t and z are random variables.
The symbols \(T\) and \(Z\) commonly represent a Student’s t random variable and a standard normal random variable. A precise connection arises when \(T\) is constructed from a standard normal \(Z\) and an independent chi-square random variable, producing the t distribution that underpins inference for a population mean when \(\sigma\) is not known.
Assumptions that link T to Z
A complete specification is required because the statement “Suppose t and z are random variables” does not fix their distributions. The standard framework in statistics assumes the following:
| Random variable | Distributional assumption | Role in the construction |
|---|---|---|
| \(Z\) | Standard normal, \(Z \sim \mathcal{N}(0,1)\) | Provides the normal numerator |
| \(U\) | Chi-square with \(\nu\) degrees of freedom, \(U \sim \chi^2_{\nu}\) | Provides the random scaling from estimating variability |
| \(T\) | \(T=\dfrac{Z}{\sqrt{U/\nu}}\) with \(Z \perp U\) | Defines a Student’s t random variable with \(\nu\) degrees of freedom |
Under these assumptions, \(T\) has a Student’s t distribution with \(\nu\) degrees of freedom, written \(T \sim t_{\nu}\). The ratio form \(Z/\sqrt{U/\nu}\) explains why the t distribution is centered like a normal distribution but has heavier tails: the denominator is random rather than fixed.
Density of the Student’s t distribution
The probability density function of \(T \sim t_{\nu}\) is
\[
f_T(t)=\frac{\Gamma\!\left(\frac{\nu+1}{2}\right)}{\sqrt{\nu\pi}\,\Gamma\!\left(\frac{\nu}{2}\right)}
\left(1+\frac{t^2}{\nu}\right)^{-\frac{\nu+1}{2}}, \qquad -\infty
Degrees of freedom \(\nu\) controls tail thickness: small \(\nu\) produces much heavier tails, and large \(\nu\) produces a curve very close to the standard normal density.
The mean exists for \(\nu>1\) and equals 0. The variance exists for \(\nu>2\) and equals
\[
\mathrm{Var}(T)=\frac{\nu}{\nu-2}.
\]
For \(\nu \le 2\), the variance is not finite, reflecting the increased probability of extreme values.
Let \(X_1,\dots,X_n\) be a random sample from a normal population \(\mathcal{N}(\mu,\sigma^2)\). The sample mean and sample standard deviation are
\[
\bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_i, \qquad
S=\sqrt{\frac{\sum_{i=1}^{n}(X_i-\bar{X})^2}{n-1}}.
\]
The classic t statistic is
\[
T=\frac{\bar{X}-\mu}{S/\sqrt{n}}.
\]
Under normality, this statistic satisfies \(T \sim t_{n-1}\).
The companion z statistic (when \(\sigma\) is known) is
\[
Z=\frac{\bar{X}-\mu}{\sigma/\sqrt{n}},
\]
and \(Z \sim \mathcal{N}(0,1)\). The difference between \(T\) and \(Z\) is the substitution of \(\sigma\) by the random estimate \(S\), which introduces the chi-square scaling and produces the t distribution.
Two qualitative comparisons are central in regression, confidence intervals, and hypothesis testing:
The t distribution statement \(T \sim t_{n-1}\) for \(\dfrac{\bar{X}-\mu}{S/\sqrt{n}}\) relies on normality of the underlying population. For non-normal populations, the t procedure is often approximately valid for large \(n\) by central limit behavior, while small-sample accuracy depends on the degree of skewness and outliers.
The notation “\(T\) and \(Z\) are random variables” also permits dependence; the construction \(T=\dfrac{Z}{\sqrt{U/\nu}}\) requires independence between \(Z\) and \(U\). Without independence, the t distribution conclusion does not generally hold.
Moments and conditions for existence
Connection to estimating a population mean when σ is not known
Comparison between the t distribution and the standard normal distribution
Feature
Standard normal \(Z\)
Student’s t \(T\)
Center and symmetry
Centered at 0, symmetric
Centered at 0, symmetric
Tail weight
Lighter tails
Heavier tails for finite \(\nu\)
Typical use
\(\sigma\) known, or large-sample approximations
\(\sigma\) unknown, especially for small to moderate \(n\)
Large-\(\nu\) behavior
Fixed
Approaches \(Z\) as \(\nu \to \infty\)
Visualization: tν and standard normal densities
Common pitfalls