The main functions to use the \(F\)-distribution in R are df(), pf(), qf() and rf(). The df() function gives the density, the pf() function gives the distribution function, the qf() function gives the quantile function and the rf() function generates random deviates.

We use the df() function to calculate the density of a \(F\)-curve with \(v_1=10\) and \(v_2=20\) at the value of 1.2:

df(1.2, df1 = 10, df2 = 20)

## [1] 0.5626125

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We use the pf() function to calculate the area under the curve for the interval \([0,1.5]\) and the interval \([1.5, +\infty)\) of a \(F\)-curve with with \(v_1=10\) and \(v_2=20\):

x <- 1.5
v1 <- 10
v2 <- 20
# interval [0,1.5]
pf(x, df = v1, df2 = v2, lower.tail = TRUE)

## [1] 0.7890535

# interval [1.5,+inf)
pf(x, df = v1, df2 = v2, lower.tail = FALSE)

## [1] 0.2109465

Further, we check if the sum of the intervals \([0,1.5]\) and \([1.5, +\infty)\) sums up to 1:

pf(x, df = v1, df2 = v2, lower.tail = TRUE) + pf(x, df = v1, df2 = v2, lower.tail = FALSE) == 1

## [1] TRUE

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We use the qf() function to calculate the quantile for a given area (= probability) under the curve, that corresponds to \(q = 0.25, 0.5, 0.75\) and \(0.999\), for a \(F\)-curve with \(v_1=10\) and \(v_2=20\). We set lower.tail = TRUE in order the get the area for the interval \([0, q]\).

q <- c(0.25, 0.5, 0.75, 0.999)
v1 <- 10
v2 <- 20
qf(q[1], df1 = v1, df2 = v2, lower.tail = TRUE)

## [1] 0.6563936

qf(q[2], df1 = v1, df2 = v2, lower.tail = TRUE)

## [1] 0.9662639

qf(q[3], df1 = v1, df2 = v2, lower.tail = TRUE)

## [1] 1.399487

qf(q[4], df1 = v1, df2 = v2, lower.tail = TRUE)

## [1] 5.075246

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Finally, we use the rf() function to generate 100,000 random values from the \(F\)-distribution with \(v_1=10\) and \(v_2=20\). Thereafter, we plot a histogram and compare it to the probability density function of the \(F\)-distribution with \(v_1=10\) and \(v_2=20\) (pink line).

x <- rf(100000, df1 = 10, df2 = 20)
hist(x,
  breaks = "Scott",
  freq = FALSE,
  xlim = c(0, 3),
  ylim = c(0, 1),
  xlab = "",
  main = (TeX("Histogram for a $\\F$-distribution with $\\v_1 = 10$ and $\\v_2 = 20$ degrees of freedom (df)")),
  cex.main = 0.9
)

curve(df(x, df1 = 10, df2 = 20), from = 0, to = 4, n = 5000, col = "pink", lwd = 2, add = T)

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Citation

The E-Learning project SOGA-R was developed at the Department of Earth Sciences by Kai Hartmann, Joachim Krois and Annette Rudolph. You can reach us via mail by soga[at]zedat.fu-berlin.de.

You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

Please cite as follow: Hartmann, K., Krois, J., Rudolph, A. (2023): Statistics and Geodata Analysis using R (SOGA-R). Department of Earth Sciences, Freie Universitaet Berlin.