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