Question

using R :-

A Gumbel random variable X has distribution function

FX (x) = exp (−e^−x).

a) Give a graph of FX and explain using this plot why FX is a valid cumulative probability distri-

bution function.

(b) Find the values of the first and third quartiles and median X and show their values on the graph.

(c) Make a table of x and FX (x) for x equal to the integers from −2 to 5.

(d) Find the probabilities P{−1 < X ≤ 4} and P{4 < X}.

(e) Find the probability density for this distribution function.

(f) Provide a second sketch of the distribution function along with a sketch of the density function indicating P {−1 < X ≤4} on both pl

Answer 1

The distribution function of the Gumbel random variable is

a) The graph of is plotted below.

R code below.

curve(exp(-exp(-x)), xlim=c(-5,7),ylim = c(0,1), lwd=2, col = "blue", xlab="x", ylab = "F(x)")
abline(v=-log(log(4)), col = "green")
abline(v=-log(log(2)), col = "green")
abline(v=-log(log(4/3)), col = "green")

We see and . and is increasing. Hence

  is a valid cumulative probability distribution function

b) First quartile is such that

Third quartile is such that

Median is such that

The green lines in the graph are 25%, 50% and 75%.

c) The table is generated below.

> matrix(c(-2:5,exp(-exp((-2:5)))), nrow=8)
[,1] [,2]
[1,] -2 8.734230e-01
[2,] -1 6.922006e-01
[3,] 0 3.678794e-01
[4,] 1 6.598804e-02
[5,] 2 6.179790e-04
[6,] 3 1.892179e-09
[7,] 4 1.942338e-24
[8,] 5 3.507389e-65

d) The probability

e) The PDF is