# Student's t-distribution Probabilities

Calculator of Student's t-distribution : probabilities, density function (PDF) and cumulative density function (CDF).

## Student's t-distribution formulas

X is a random variable following a Student's t-distribution of parameter `k > 0` (degrees of freedom).

`Gamma(x)` is the gamma function.

### Probability Density Function (PDF)

`f(x) = 1/sqrt(k*pi)*(Gamma((k+1)/2))/(Gamma(k/2))*(1+x^2/k)^-((k+1)/2)`

### Cumulative distribution function (CDF)

`F(x) = \int_-oo^x f(t)\ dt`

`F(x) = 1/2+x*Gamma((k+1)/2)* (\text{}_2F_1(1/2,(k+1)/2;3/2;-x^2/k))/(sqrt(k*pi)*Gamma(k/2))`

where `\text{}_2F_1` is the hypergeometric function.

### Probabilities

Probability that X is greater than a :

`P(X > a) = 1 - F(a)`

Probability that X is less than b :

`P(X < b) = F(b)`

Probability that X lies between a and b :

`P(a < X < b) = F(b) - F(a)`

## See also

Inverse Student's t-distribution

Student's t-distribution Chart

Statistics Calculators