Here, we discuss Student’s t distribution functions in R, plots, parameter setting, random sampling, density, cumulative distribution and quantiles.

The Student’s t distribution with parameter \(\tt{degree\;of\;freedom}=\nu\) has probability density function (pdf) formula as:

\[\begin{align} f(x)& =\frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\,\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}} \\ & = \frac{1}{\sqrt{\nu}\,\mathrm{B} (\frac{1}{2}, \frac{\nu}{2})} \left(1+\frac{x^2}\nu \right)^{-\frac{\nu+1}{2}},\end{align}\] for \(x\in\mathbb{R}\), where \(\nu > 0\),

\(\Gamma\) is the \(\tt{gamma\;function}\), and \(\mathrm{B}\) is the \(\tt{beta\;function}\).

The mean is \(0\) for \(\nu > 1\), otherwise undefined,

the variance is \(\frac{\nu}{\nu-2}\) for \(\nu > 2\); \(\infty\) for \(1 < \nu \le 2\); and otherwise undefined.

See also probability distributions and plots and charts.

1 Table of Student’s t Distribution Functions in R

The table below shows the functions for Student’s t distributions in R.

Table of Student’s t Distribution Functions in R
Function Usage
rt(n, df, ncp) Simulate a random sample with \(n\) observations
dt(x, df, ncp) Calculate the probability density at the point \(x\)
pt(q, df, ncp) Calculate the cumulative distribution at the point \(q\)
qt(p, df, ncp) Calculate the quantile value associated with \(p\)

The examples here are central Student’s t distributions, hence, the "ncp" argument is excluded in the examples below.

However, for non-central Student’s t distributions, you can set the argument of the non-centrality parameter value to a non-zero value as ncp = 0 is central. For example:

dt(2.12, df = 25, ncp = 0)
[1] 0.0460441
# It is the same as:
dt(2.12, df = 25)
[1] 0.0460441

2 Plot of Student’s t Distributions in R

Single distribution:

Below is a plot of the Student’s t distribution function with \(\tt{degree\;of\;freedom\;}=35\).

x = seq(-4, 4, 1/1000); y = dt(x, 35)
plot(x, y, type = "l",
     xlim = c(-4, 4), ylim = c(0, max(y)),
     main = "Probability Density Function of Student's t Distribution (35)",
     xlab = "x", ylab = "Density",
     lwd = 2, col = "blue")
# Add line and legend
lines(c(0, 0), c(0, max(y)), col = "red")
legend("topleft", "df = 35",
       lwd = 2,
       col = "blue",
       bty = "n")
Probability Density Function (PDF) of a Student's t Distribution in R

Probability Density Function (PDF) of a Student’s t Distribution in R

Multiple distributions:

Below is a plot of multiple Student’s t distribution functions in one graph.

x1 = seq(-7, 7, 1/1000); y1 = dt(x1, 60)
x2 = seq(-7, 7, 1/1000); y2 = dt(x2, 5)
x3 = seq(-7, 7, 1/1000); y3 = dt(x3, 2)
plot(x1, y1, type = "l",
     xlim = c(-7, 7), ylim = range(c(y1, y2, y3)),
     main = "Probability Density Functions of Student's t Distributions",
     xlab = "x", ylab = "Density",
     lwd = 2, col = "blue")
points(x2, y2, type = "l", lwd = 2, col = "red")
points(x3, y3, type = "l", lwd = 2, col = "green")
# Add legend
legend("topleft", c("df = 60",
                    "df = 5",
                    "df = 2"),
       lwd = c(2, 2, 2),
       col = c("blue", "red", "green"),
       bty = "n")
Probability Density Functions (PDFs) of Student's t Distributions in R

Probability Density Functions (PDFs) of Student’s t Distributions in R

3 Examples for Setting Parameters for Student’s t Distributions in R

To set the parameter for the Student’s t distribution function, with \(\tt{degrees\;of\;freedom}=15\).

For example, for pt(), the following are the same:

pt(2.2, 15)
[1] 0.9780522
pt(2.2, df = 15)
[1] 0.9780522

4 rt(): Random Sampling from Student’s t Distributions in R

Sample 1500 observations from the Student’s t distribution with \(\tt{degree\;of\;freedom\;}=60\):

rt(1500, 60)
set.seed(100) # Line allows replication (use any number).
sample = rt(1500, 60)
hist(sample,
     main = "Histogram of 1500 Observations from Student's t
Distribution with 60 Degrees of Freedom",
     xlab = "x",
     col = "deepskyblue", border = "white")
Histogram of Student's t Distribution (60) Random Sample in R

Histogram of Student’s t Distribution (60) Random Sample in R

5 dt(): Probability Density Function for Student’s t Distributions in R

Calculate the density at \(x = 1.5\), in the Student’s t distribution with with \(\tt{degree\;of\;freedom\;}=40\):

dt(1.5, 40)
[1] 0.1291154
x = seq(-4, 4, 1/1000); y = dt(x, 40)
plot(x, y, type = "l",
     xlim = c(-4, 4), ylim = c(0, max(y)),
     main = "Probability Density Function of Student's t Distribution
with 40 Degrees of Freedom",
     xlab = "x", ylab = "Density",
     lwd = 2, col = "blue")
# Add lines
segments(1.5, -1, 1.5, 0.1291154)
segments(-5, 0.1291154, 1.5, 0.1291154)
Probability Density Function (PDF) of Student's t Distribution (40) in R

Probability Density Function (PDF) of Student’s t Distribution (40) in R

6 pt(): Cumulative Distribution Function for Student’s t Distributions in R

Calculate the cumulative distribution at \(x = 1.6\), in the Student’s t distribution with \(\tt{degree\;of\;freedom\;}=80\). That is, \(P(X \le 1.6)\):

pt(1.6, 80)
[1] 0.9432297
x = seq(-4, 4, 1/1000); y = pt(x, 80)
plot(x, y, type = "l",
     xlim = c(-4, 4), ylim = c(0,1),
     main = "Cumulative Distribution Function of Student's t Distribution
with 80 Degrees of Freedom",
     xlab = "x", ylab = "Cumulative Distribution",
     lwd = 2, col = "blue")
# Add lines
segments(1.6, -1, 1.6, 0.9432297)
segments(-5, 0.9432297, 1.6, 0.9432297)
Cumulative Distribution Function (CDF) of Student's t Distribution (80) in R

Cumulative Distribution Function (CDF) of Student’s t Distribution (80) in R 

x = seq(-4, 4, 1/1000); y = dt(x, 80)
plot(x, y, type = "l",
     xlim = c(-4, 4), ylim = c(0, max(y)),
     main = "Probability Density Function of Student's t Distribution
with 80 Degrees of Freedom",
     xlab = "x", ylab = "Density",
     lwd = 2, col = "blue")
# Add shaded region and legend
point = 1.6
polygon(x = c(x[x <= point], point),
        y = c(y[x <= point], 0),
        col = "limegreen")
legend("topright", c("Area = 0.9432297"),
       fill = c("limegreen"),
       inset = 0.01)
Shaded Probability Density Function (PDF) of Student's t Distribution (80) in R

Shaded Probability Density Function (PDF) of Student’s t Distribution (80) in R

For upper tail, at \(x = 1.6\), that is, \(P(X \ge 1.6) = 1 - P(X \le 1.6)\), set the "lower.tail" argument:

pt(1.6, 80, lower.tail = FALSE)
[1] 0.05677031
x = seq(-4, 4, 1/1000); y = dt(x, 80)
plot(x, y, type = "l",
     xlim = c(-4, 4), ylim = c(0, max(y)),
     main = "Shaded Upper Region: Probability Density Function of
Student's t Distribution with 80 Degrees of Freedom",
     xlab = "x", ylab = "Density",
     lwd = 2, col = "blue")
# Add shaded region and legend
point = 1.6
polygon(x = c(point, x[x >= point]),
        y = c(0, y[x >= point]),
        col = "limegreen")
legend("topright", c("Area = 0.05677031"),
       fill = c("limegreen"),
       inset = 0.01)
Shaded Upper Region: Probability Density Function (PDF) of Student's t Distribution (80) in R

Shaded Upper Region: Probability Density Function (PDF) of Student’s t Distribution (80) in R

7 qt(): Derive Quantile for Student’s t Distributions in R

Derive the quantile for \(p = 0.9\), in the Student’s t distribution with \(\tt{degree\;of\;freedom\;}=50\). That is, \(x\) such that, \(P(X \le x)=0.9\):

qt(0.9, 50)
[1] 1.298714
x = seq(-4, 4, 1/1000); y = pt(x, 50)
plot(x, y, type = "l",
     xlim = c(-4, 4), ylim = c(0,1),
     main = "Cumulative Distribution Function of Student's t Distribution
with 50 Degrees of Freedom",
     xlab = "x", ylab = "Cumulative Distribution",
     lwd = 2, col = "blue")
# Add lines
segments(1.298714, -1, 1.298714, 0.9)
segments(-5, 0.9, 1.298714, 0.9)
Cumulative Distribution Function (CDF) of Student's t Distribution (50) in R

Cumulative Distribution Function (CDF) of Student’s t Distribution (50) in R 

x = seq(-4, 4, 1/1000); y = dt(x, 50)
plot(x, y, type = "l",
     xlim = c(-4, 4), ylim = c(0, max(y)),
     main = "Probability Density Function of Student's t Distribution
with 50 Degrees of Freedom",
     xlab = "x", ylab = "Density",
     lwd = 2, col = "blue")
# Add shaded region and legend
point = 1.298714
polygon(x = c(x[x <= point], point),
        y = c(y[x <= point], 0),
        col = "limegreen")
legend("topright", c("Area = 0.9"),
       fill = c("limegreen"),
       inset = 0.01)
Shaded Probability Density Function (PDF) of Student's t Distribution (50) in R

Shaded Probability Density Function (PDF) of Student’s t Distribution (50) in R

For upper tail, for \(p = 0.1\), that is, \(x\) such that, \(P(X \ge x)=0.1\):

qt(0.1, 50, lower.tail = FALSE)
[1] 1.298714
x = seq(-4, 4, 1/1000); y = dt(x, 50)
plot(x, y, type = "l",
     xlim = c(-4, 4), ylim = c(0, max(y)),
     main = "Shaded Upper Region: Probability Density Function of
Student's t Distribution with 50 Degrees of Freedom",
     xlab = "x", ylab = "Density",
     lwd = 2, col = "blue")
# Add shaded region and legend
point = 1.298714
polygon(x = c(point, x[x >= point]),
        y = c(0, y[x >= point]),
        col = "limegreen")
legend("topright", c("Area = 0.1"),
       fill = c("limegreen"),
       inset = 0.01)
Shaded Upper Region: Probability Density Function (PDF) of Student's t Distribution (50) in R

Shaded Upper Region: Probability Density Function (PDF) of Student’s t Distribution (50) in R

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