Student's t-distribution

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Release:

4.6  •  5.0  •  5.1  •  5.2

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.

StudentT(5).png

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

$ 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 $ d $ is «dof», and $ \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».

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

Example

Student-T.jpg

See Also

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