Evaluate the probability mass function of a StudentsT distribution
Please see the documentation of StudentsT() for some properties
of the StudentsT distribution, as well as extensive examples
showing to how calculate p-values and confidence intervals.
## S3 method for class 'StudentsT' pdf(d, x, ...) ## S3 method for class 'StudentsT' log_pdf(d, x, ...)
d |
A |
x |
A vector of elements whose probabilities you would like to
determine given the distribution |
... |
Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors. |
A vector of probabilities, one for each element of x.
Other StudentsT distribution: cdf.StudentsT,
quantile.StudentsT,
random.StudentsT
set.seed(27) X <- StudentsT(3) X random(X, 10) pdf(X, 2) log_pdf(X, 2) cdf(X, 4) quantile(X, 0.7) ### example: calculating p-values for two-sided T-test # here the null hypothesis is H_0: mu = 3 # data to test x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2) nx <- length(x) # calculate the T-statistic t_stat <- (mean(x) - 3) / (sd(x) / sqrt(nx)) t_stat # null distribution of statistic depends on sample size! T <- StudentsT(df = nx - 1) # calculate the two-sided p-value 1 - cdf(T, abs(t_stat)) + cdf(T, -abs(t_stat)) # exactly equivalent to the above 2 * cdf(T, -abs(t_stat)) # p-value for one-sided test # H_0: mu <= 3 vs H_A: mu > 3 1 - cdf(T, t_stat) # p-value for one-sided test # H_0: mu >= 3 vs H_A: mu < 3 cdf(T, t_stat) ### example: calculating a 88 percent T CI for a mean # lower-bound mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx) # upper-bound mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx) # equivalent to mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx) # also equivalent to mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx) mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)
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