Student's t-Distribution
A Student's t-Distribution is a generalised hyperbolic distribution of the ratio [math]\displaystyle{ \frac{Z}{\sqrt{V/\nu}} = (Z+\mu) \sqrt{\frac{\nu}{V}} }[/math], where Z is normally distributed (with expected value 0 and variance of 1); V has a chi-squared distribution with [math]\displaystyle{ \nu }[/math] degrees of freedom; Z and [math]\displaystyle{ V }[/math] are independent; and μ is the Noncentrality Parameter.
- AKA: t-Distribution.
- Context:
- The distribution can arises in estimating the expected value of a normally distributed population when the sample size is small.
- It can be used by a Student's t-Test.
- It can be instantiated in a t-Statistic (to produce a t-score).
- …
- Counter-Example(s):
- See: Z-Distribution, R. A. Fisher.
References
2009
- (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Student%27s_t-distribution
- QUOTE: In probability and statistics, 'Student’s t-distribution (or simply the t-distribution) is a continuous probability distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown. It plays a role in a number of widely-used statistical analyses, including the Student’s t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in [[linear
regression analysis]]. The Student’s t-distribution also arises in the Bayesian analysis of data from a normal family.
The t-distribution is symmetric and bell-shaped, like the normal distribution, but has heavier tails, meaning that it is more prone to producing values that fall far from its mean. This makes it useful for understanding the statistical behavior of certain types of ratios of random quantities, in which variation in the denominator is amplified and may produce outlying values when the denominator of the ratio falls close to zero. The Student’s t-distribution is a special case of the generalised hyperbolic distribution.
- http://en.wikipedia.org/wiki/Student%27s_t-distribution#Definition
- QUOTE: Student's t-distribution has the probability density function given by [math]\displaystyle{ f(t) = \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{t^2}{\nu} \right)^{-\frac{\nu+1}{2}},\! }[/math] where [math]\displaystyle{ \nu }[/math] is the number of degrees of freedom and [math]\displaystyle{ \Gamma }[/math] is the Gamma function. This may also be written as [math]\displaystyle{ f(t) = \frac{1}{\sqrt{\nu}\, B \left (\frac{1}{2}, \frac{\nu}{2}\right )} \left(1+\frac{t^2}{\nu} \right)^{-\frac{\nu+1}{2}}\!, }[/math] where B is the Beta function.
For [math]\displaystyle{ \nu }[/math] even, [math]\displaystyle{ \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} = \frac{(\nu -1)(\nu -3)\cdots 5 \cdot 3} {2\sqrt{\nu}(\nu -2)(\nu -4)\cdots 4 \cdot 2\,}. }[/math]
For [math]\displaystyle{ \nu }[/math] odd, [math]\displaystyle{ \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} = \frac{(\nu -1)(\nu -3)\cdots 4 \cdot 2} {\pi \sqrt{\nu}(\nu -2)(\nu -4)\cdots 5 \cdot 3\,}.\! }[/math]
The overall shape of the probability density function of the t-distribution resembles the bell shape of a normally distributed variable with mean 0 and variance 1, except that it is a bit lower and wider. As the number of degrees of freedom grows, the t-distribution approaches the normal distribution with mean 0 and variance 1.
- QUOTE: Student's t-distribution has the probability density function given by [math]\displaystyle{ f(t) = \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{t^2}{\nu} \right)^{-\frac{\nu+1}{2}},\! }[/math] where [math]\displaystyle{ \nu }[/math] is the number of degrees of freedom and [math]\displaystyle{ \Gamma }[/math] is the Gamma function. This may also be written as [math]\displaystyle{ f(t) = \frac{1}{\sqrt{\nu}\, B \left (\frac{1}{2}, \frac{\nu}{2}\right )} \left(1+\frac{t^2}{\nu} \right)^{-\frac{\nu+1}{2}}\!, }[/math] where B is the Beta function.
1995
- (Johnson et al., 1995) ⇒ Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). “Continuous Univariate Distributions, Volume 2, 2nd Edition.” Wiley, ISBN 0-471-58494-0
1954
- (Dunnet & Sobel, 1954) ⇒ C. W. Dunnett, and M. Sobel. (1954). “A bivariate generalization of Student's t-distribution, with tables for certain special cases.” In: Biometrika, 41(1-2).