tcdf

tcdf is the name commonly used for the cumulative distribution function of the Student's t-distribution. It returns the probability that a t-distributed random variable with a given number of degrees of freedom is less than or equal to a specified value.

Parameters and domain: The function takes x, the value at which to evaluate the CDF, and df (degrees of freedom), a positive real number. The CDF is defined for all real x when df > 0. In many implementations, the CDF can be computed by numerical integration of the t-distribution’s probability density function or by a closed-form expression using the regularized incomplete beta function, with z = df / (df + x^2).

Formula and interpretation: A commonly used closed-form representation is F(x; df) = 0.5 * I_z(df/2, 0.5) for x >= 0, and F(x; df) = 1 − 0.5 * I_z(df/2, 0.5) for x < 0, where z = df / (df + x^2) and I_z(a,b) is the regularized incomplete beta function. This yields F(0; df) = 0.5 and, as df grows, the t-distribution approaches the standard normal distribution with F(x; df) converging to Φ(x).

Usage and connections: tcdf is used to compute p-values for t-tests and to obtain CDF values for observed t-statistics. It connects to the normal CDF in the large-df limit and to other t-distribution functions for noncentral cases.

Software implementations: Implementations include MATLAB's tcdf(x, df) and SciPy's scipy.stats.t.cdf(x, df). Other environments provide equivalent functions (for example, R users may use pt with appropriate tail settings).