Student's t Distribution Formula

t Distribution Formula

We wish to consider the formula that is used to define all t-distributions.  It is easy to see from the formula above that there are many ingredients that go into making a t-distribution.  This formula is actually a composition of many types of functions.  A few items in the formula need a little explanation.

• The symbol Γ is the capital form of the Greek letter gamma.  This refers to the gamma function.  The gamma function is defined in a complicated way using calculus and is a generalization of the factorial.
• The symbol ν is the Greek lower case letter nu and refers to the number of degrees of freedom of the distribution.
• The symbol π is the Greek lower case letter pi and is the mathematical constant that is approximately 3.14159. . .

There are many features about the graph of the probability density function that can be seen as a direct consequence of this formula.

• These types of distributions are symmetric about the y-axis.  The reason for this has to do with the form of the function defining our distribution.  This function is an even function, and even functions display this type of symmetry.  As a consequence of this symmetry, the mean and the median coincide for every t-distribution.
• There is a horizontal asymptote y = 0 for the graph of the function.  We can see this if we calculate limits at infinity.  Due to the negative exponent, as t  increases or decreases without bound, the function approaches zero.
• The function is nonnegative.  This is a requirement for all probability density functions.

Other features require a more sophisticated analysis of the function.  These features include the following:

• The graphs of t distributions are bell-shaped, but are not normally distributed.
• The tails of a t distribution are thicker than what the tails of the normal distribution are.
• Every t distribution has a single peak.
• As the number of degrees of freedom increase, the corresponding t distributions become more and more normal in appearance.  The standard normal distribution is the limit of this process. 

Using a Table Instead of the Formula

The function that defines a t distribution is quite complicated to work with.  Many of the above statements require some topics from calculus to demonstrate.  Fortunately, most of the time we do not need to use the formula.  Unless we are attempting to prove a mathematical result about the distribution, it is usually easier to deal with a table of values.  A table such as this has been developed using the formula for the distribution.  With the proper table, we do not need to work directly with the formula.