1.3. EDA Techniques
1.3.6. Probability Distributions
1.3.6.6. Gallery of Distributions
1.3.6.6.4. |
t Distribution |
![f(x) = (1 + x**2/nu)**(-(nu+1)/2)/[B(0.5,0.5*nu)*SQRT(nu)]](eqns/tpdf.gif)
where 
is the beta function and
is a positive integer shape parameter.
The formula for the beta function is
In a testing context, the t distribution is treated as a
"standardized distribution" (i.e., no location or scale parameters).
However, in a distributional modeling context (as with other
probability distributions), the t distribution itself can be
transformed with a location parameter,
, and a
scale parameter,
.
The following is the plot of the t probability density function for 4 different values of the shape parameter.
These plots all have a similar shape. The difference is in the
heaviness of the tails. In fact, the t distribution with
equal to 1 is a
Cauchy distribution. The t
distribution approaches a normal distribution
as becomes large. The approximation
is quite good for values of > 30.
The following are the plots of the t cumulative distribution
function with the same values of as
the pdf plots above.
The following are the plots of the t percent point function
with the same values of as the pdf
plots above.
| Mean |
0 (It is undefined for equal
to 1.)
|
| Median | 0 |
| Mode | 0 |
| Range | Infinity in both directions. |
| Standard Deviation |
It is undefined for
|
| Coefficient of Variation | Undefined |
| Skewness |
0. It is undefined for
less than or equal to 3.
However, the t distribution is symmetric in all cases.
|
| Kurtosis |
It is undefined for
|
