1 Table of F Distribution Functions in R
2 Plot of F Distributions in R
3 Examples for Setting Parameters for F Distributions in R
4 rf(): Random Sampling from F Distributions in R
5 df(): Probability Density Function for F Distributions in R
6 pf(): Cumulative Distribution Function for F Distributions in R
7 qf(): Derive Quantile for F Distributions in R

Here, we discuss F distribution functions in R, plots, parameter setting, random sampling, density, cumulative distribution and quantiles.

The F distribution with parameters \(\tt{degrees\;of\;freedom\;1}=d_1\), and \(\tt{degrees\;of\;freedom\;2}=d_2\) has probability density function (pdf) formula as:

\[\begin{align} f(x)& =\frac{\Gamma \left(\frac{d_1 +d_2}{2} \right)} {\Gamma \left(\frac{d_1}{2} \right)\,\Gamma \left(\frac{d_2}{2} \right)} \left(\frac{d_1}{d_2}\right)^{d_1/2} x^{d_1/2 - 1} \left(1+\frac{d_1}{d_2} \, x \right)^{-(d_1+d_2)/2}\\ & = \frac{1}{\operatorname{B}\left(\frac{d_1}{2},\frac{d_2}{2}\right)} \left(\frac{d_1}{d_2}\right)^{d_1/2} x^{d_1/2 - 1} \left(1+\frac{d_1}{d_2} \, x \right)^{-(d_1+d_2)/2},\end{align}\]

for \(x \in (0, +\infty)\), if \(d_1 = 1\),

otherwise \(x \in [0, +\infty)\), where \(d_1>0\), and \(d_2>0\),

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

The mean is \(\frac{d_2}{d_2-2}\) for \(d_2 > 2\), and variance is \(\frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\) for \(d_2 > 4\).

1 Table of F Distribution Functions in R

The table below shows the functions for F distributions in R.

Table of F Distribution Functions in R
Function	Usage
rf(n, df1, df2, ncp)	Simulate a random sample with \(n\) observations
df(x, df1, df2, ncp)	Calculate the probability density at the point \(x\)
pf(q, df1, df2, ncp)	Calculate the cumulative distribution at the point \(q\)
qf(p, df1, df2, ncp)	Calculate the quantile value associated with \(p\)

The examples here are central F distributions, hence, the "ncp" argument is excluded in the examples below.

However, for non-central F distributions, you can set the argument of the non-centrality parameter value to a non-zero value as ncp = 0 is central. For example:

pf(5.3, df1 = 2, df2 = 12, ncp = 0)

[1] 0.9775903

# It is the same as:
pf(5.3, df1 = 2, df2 = 12)

[1] 0.9775903

2 Plot of F Distributions in R

Single distribution:

Below is a plot of the F distribution function with \(\tt{df1}=3\) and \(\tt{df2}=30\).

x = seq(0, 6, 1/1000); y = df(x, 3, 30)
plot(x, y, type = "l",
     xlim = c(0, 6), ylim = c(0, max(y)),
     main = "Probability Density Function of F Distribution (3, 30)",
     xlab = "x", ylab = "Density",
     lwd = 2, col = "blue")
# Add legend
legend("topright", "df1 = 3, df2 = 30",
       lwd = 2,
       col = "blue",
       bty = "n")

Probability Density Function (PDF) of an F Distribution in R

Multiple distributions:

Below is a plot of multiple F distribution functions in one graph.

x1 = seq(0, 10, 1/1000); y1 = df(x1, 2, 50)
x2 = seq(0, 10, 1/1000); y2 = df(x2, 20, 20)
x3 = seq(0, 10, 1/1000); y3 = df(x3, 50, 2)
plot(x1, y1, type = "l",
     xlim = c(0, 10), ylim = range(c(y1, y2, y3)),
     main = "Probability Density Functions of F 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("topright", c("df1 = 2,   df2 = 50",
                    "df1 = 20, df2 = 20",
                    "df1 = 50, df2 = 2"),
       lwd = c(2, 2, 2),
       col = c("blue", "red", "green"),
       bty = "n")

Probability Density Functions (PDFs) of F Distributions in R

3 Examples for Setting Parameters for F Distributions in R

To set the parameters for the F distribution function, with \(\tt{df1} = 4\) and \(\tt{df2} = 80\).

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

# The order of 4 and 80 matters here as the parameter names are not used.
# The first number 4 is df1, and 80 is df2.
qf(0.5, 4, 80)

[1] 0.846329

qf(0.5, df1 = 4, df2 = 80); qf(0.5, df2 = 80, df1 = 4)

[1] 0.846329

[1] 0.846329

4 rf(): Random Sampling from F Distributions in R

Sample 900 observations from the F distribution with \(\tt{df1} = 2\) and \(\tt{df2} = 60\):

rf(900, 2, 60)

set.seed(12) # Line allows replication (use any number).
sample = rf(900, 2, 60)
hist(sample,
     main = "Histogram of 900 Observations from F Distribution
with Df1 = 2 and Df2 = 60",
     xlab = "x",
     col = "deepskyblue", border = "white")

Histogram of F Distribution (2, 60) Random Sample in R

5 df(): Probability Density Function for F Distributions in R

Calculate the density at \(x = 1.6\), in the F distribution with \(\tt{df1} = 6\) and \(\tt{df2} = 45\):

df(1.6, 6, 45)

[1] 0.2837588

x = seq(0, 5, 1/1000); y = df(x, 6, 45)
plot(x, y, type = "l",
     xlim = c(0, 5), ylim = c(0, max(y)),
     main = "Probability Density Function of F Distribution
with Df1 = 6 and Df2 = 45",
     xlab = "x", ylab = "Density",
     lwd = 2, col = "blue")
# Add lines
segments(1.6, -1, 1.6, 0.2837588)
segments(-1, 0.2837588, 1.6, 0.2837588)

Probability Density Function (PDF) of F Distribution (6, 45) in R

6 pf(): Cumulative Distribution Function for F Distributions in R

Calculate the cumulative distribution at \(x = 2.1\), in the F distribution with \(\tt{df1} = 7\) and \(\tt{df2} = 40\). That is, \(P(X \le 2.1)\):

pf(2.1, 7, 40)

[1] 0.93415

x = seq(0, 4, 1/1000); y = pf(x, 7, 40)
plot(x, y, type = "l",
     xlim = c(0, 4), ylim = c(0,1),
     main = "Cumulative Distribution Function of F Distribution
with Df1 = 7 and Df2 = 40",
     xlab = "x", ylab = "Cumulative Distribution",
     lwd = 2, col = "blue")
# Add lines
segments(2.1, -1, 2.1, 0.93415)
segments(-1, 0.93415, 2.1, 0.93415)

Cumulative Distribution Function (CDF) of F Distribution (7, 40) in R

x = seq(0, 4, 1/1000); y = df(x, 7, 40)
plot(x, y, type = "l",
     xlim = c(0, 4), ylim = c(0, max(y)),
     main = "Probability Density Function of F Distribution
with Df1 = 7 and Df2 = 40",
     xlab = "x", ylab = "Density",
     lwd = 2, col = "blue")
# Add shaded region and legend
point = 2.1
polygon(x = c(x[x <= point], point),
        y = c(y[x <= point], 0),
        col = "limegreen")
legend("topright", c("Area = 0.93415"),
       fill = c("limegreen"),
       inset = 0.01)

Shaded Probability Density Function (PDF) of F Distribution (7, 40) in R

For upper tail, at \(x = 2.1\), that is, \(P(X \ge 2.1) = 1 - P(X \le 2.1)\), set the "lower.tail" argument:

pf(2.1, 7, 40, lower.tail = FALSE)

[1] 0.06584997

x = seq(0, 4, 1/1000); y = df(x, 7, 40)
plot(x, y, type = "l",
     xlim = c(0, 4), ylim = c(0, max(y)),
     main = "Shaded Upper Region: Probability Density Function of
F Distribution with Df1 = 7 and Df2 = 40",
     xlab = "x", ylab = "Density",
     lwd = 2, col = "blue")
# Add shaded region and legend
point = 2.1
polygon(x = c(point, x[x >= point]),
        y = c(0, y[x >= point]),
        col = "limegreen")
legend("topright", c("Area = 0.06584997"),
       fill = c("limegreen"),
       inset = 0.01)

Shaded Upper Region: Probability Density Function (PDF) of F Distribution (7, 40) in R

7 qf(): Derive Quantile for F Distributions in R

Derive the quantile for \(p = 0.85\), in the F distribution with \(\tt{df1} = 6\) and \(\tt{df2} = 35\). That is, \(x\) such that, \(P(X \le x)=0.85\):

qf(0.85, 6, 35)

[1] 1.700715

x = seq(0, 4, 1/1000); y = pf(x, 6, 35)
plot(x, y, type = "l",
     xlim = c(0, 4), ylim = c(0,1),
     main = "Cumulative Distribution Function of
F Distribution with Df1 = 6 and Df2 = 35",
     xlab = "x", ylab = "Cumulative Distribution",
     lwd = 2, col = "blue")
# Add lines
segments(1.700715, -1, 1.700715, 0.85)
segments(-1, 0.85, 1.700715, 0.85)

Cumulative Distribution Function (CDF) of F Distribution (6, 35) in R

x = seq(0, 4, 1/1000); y = df(x, 6, 35)
plot(x, y, type = "l",
     xlim = c(0, 4), ylim = c(0, max(y)),
     main = "Probability Density Function of F Distribution
with Df1 = 6 and Df2 = 35",
     xlab = "x", ylab = "Density",
     lwd = 2, col = "blue")
# Add shaded region and legend
point = 1.700715
polygon(x = c(x[x <= point], point),
        y = c(y[x <= point], 0),
        col = "limegreen")
legend("topright", c("Area = 0.85"),
       fill = c("limegreen"),
       inset = 0.01)

Shaded Probability Density Function (PDF) of F Distribution (6, 35) in R

For upper tail, for \(p = 0.15\), that is, \(x\) such that, \(P(X \ge x)=0.15\):

qf(0.15, 6, 35, lower.tail = FALSE)

[1] 1.700715

x = seq(0, 4, 1/1000); y = df(x, 6, 35)
plot(x, y, type = "l",
     xlim = c(0, 4), ylim = c(0, max(y)),
     main = "Shaded Upper Region: Probability Density Function of
F Distribution with Df1 = 6 and Df2 = 35",
     xlab = "x", ylab = "Density",
     lwd = 2, col = "blue")
# Add shaded region and legend
point = 1.700715
polygon(x = c(point, x[x >= point]),
        y = c(0, y[x >= point]),
        col = "limegreen")
legend("topright", c("Area = 0.15"),
       fill = c("limegreen"),
       inset = 0.01)

Shaded Upper Region: Probability Density Function (PDF) of F Distribution (6, 35) in R