# Student's t-distribution

Release: 4.6  •  5.0  •  5.1  •  5.2  •  5.3  •  5.4  •  6.0  •  6.1  •  6.2  •  6.3

The Student's t-distribution describes the deviation of a sample mean from the true mean when the samples are generated by a normally distributed process. It is a continuous, unbounded, symmetric and unimodal distribution.

The statistic

t = (m - u)/(s*Sqrt(n))

where m is the sample mean, u the actual mean, s the sample standard deviation, and n the sample size, is distributed according to the Student's t-distribution with n - 1 degrees of freedom. The parameter, «dof», is the degrees of freedom. Student's t-distributions are bell-shaped, much like a normal distribution, but with heavier tails, especially for smaller degrees of freedom. When n = 1, it is known as the Cauchy distribution. For efficiency reasons, when a latin-hypercube sampling method is selected, psuedo-latin-hypercube method is used to sample the Student-T, which samples from the T-distribution, but does not guarantee a perfect latin spread of the samples.

## Functions

### StudentT( dof, over )

The distribution function. Use this to specify that a chance variable or uncertain quantity has a Student's t-distribution with «dof» degrees of freedom.

Use the optional «over» parameter to create independent and identically distributed quantities over one or more indexes.

### DensStudentT(x, dof, over)

The probability density at «x», given by

$\displaystyle{ p(x) = { {\Gamma\left({ {d+1}\over 2}\right)}\over {\sqrt{\pi d} \Gamma\left( d\over 2 \right) }} \left( 1 + { x^2 \over d} \right)^{-{ {d+1}\over 2} } }$

where $\displaystyle{ d }$ is «dof», and $\displaystyle{ \Gamma(x) }$ is GammaFn.

### CumStudentT(x, dof, over)

The cumulative density up to «x», i.e., the probability that the outcome is less than or equal to «x».

$\displaystyle{ F(x) = { {\Gamma\left({ {d+1}\over 2}\right)}\over {\sqrt{\pi d} \Gamma\left( d\over 2 \right) }} \int_{\infty}^x \left( 1 + { t^2 \over d} \right)^{-{ {d+1}\over 2} } dt }$

### CumStudentTInv(p, dof, over)

The inverse cumulative probability function, aka quantile function. This is value x at which the area under the probability density graph falling at or to the left of x is «p».

## Statistics

When 0<dof<=1, all moments are undefined.

The theoretical statistics (i.e., in the absence of sampling error) when dof>1 are as follows.

• Mean = Mode = Median = 0
• Variance = $\displaystyle{ \left\{\begin{array}{ll} \infty & \mbox{when } 1 \lt dof \leq 2 \\ dof / (dof-2) & \mbox{when } dof\gt 2\end{array}\right. }$
• Skewness = 0, when dof>3.
• Kurtosis = $\displaystyle{ \left\{\begin{array}{ll} \infty & \mbox{when } 2 \lt dof \leq 4 \\ 6 / (dof-4) & \mbox{when } dof\gt 4\end{array}\right. }$

## Parameter Estimation

If you want to estimate the parameter from sample data X indexed by I, you can use the following estimation formula provided that Variance(X, I) > 1:

«dof» := 2*Variance(X, I)/(Variance(X, I) - 1)