The Student's T Distribution

The Student's t distribution

Background

The Student's t distribution, which is also called the t distribution, is a continuous probability that used to estimate the population parameters when the sample size is small or when the population standard variance is unknown.

The density function with the parameter n:

The function of the t distribution:

And (n)= ∫_0^(+∞)▒μ^(n/2-1) e^(-μ) □(24&du) which is the Euler's gamma.

Expectation and Variance

For a continuous distribution, the expectation is , so the expectation of the student t distribution is

So if X~t(n) then E(X)=0 for when n=1, the expectation dose not exist, the variance does not exist.

When 1<n<3, E(X)=0, Var(X) does not exist

When n≥3, since E(X)=0, Var(X) = E(X2) =

=

Useful properties

Comparing with normal distribution, the shapes of the t distribution is bell shaped which reflects a smaller sample size. Also, the shape of the t distribution depends on the degree of freedom. As the sample size increases, the t distribution approaches to standard normal distribution. And the t distribution has a greater dispersion than normal distribution. In addition, the expectation of the t distribution is 0 because of symmetry. Also, the variance>1 always exist when n≥3, though it is close to 1 when there are many degree of freedom.

Applications in Finance

The T value (t= (× ̅-μ)/(s/√n)) with the degree of freedom n-1 is used when the population standard derivation is unknown.

The t distribution is an important method to illustrate the interval estimation of a population mean when the population standard derivation is unknown. The margin of error is given t_(α/2) s/√n where s is the sample standard deviation, and t provides an area of α/2 in the upper tail of the t distribution with a degree of freedom is n-1. × ̅±t_(α/2) s/√n is the confidence intervals, which often used when the sample size is small (maybe less than 30) or σ is unknown. Because when the sample size is small orσ is unknown, using Z score of confidence interval is inappropriate. Companies or the whole industry always use the t distribution to estimate some of their products' percent of pass or failed or to test whether the product is in an interval of their customers' needs. For example, an automobile