Q.6.11 - A First Course in Probability

Question

Let X1, X2, X3, X4, X5 be independent continuous random variables having a common distribution function F and density function f, and set I = P{X1 < X2 < X3 < X4 < X5} 

(a) Show that I does not depend on F. Hint: Write I as a five-dimensional integral and make the change of variables ui = F(xi), i = 1, ... , 5. 

(b) Evaluate I. 

(c) Give an intuitive explanation for your answer to (b). 

Step-by-Step Solution

Answer

The required probability is $I=\frac{1}{5!}$

Step 1: Explanation (part a)

Using the definition of probability measure of random vector, we have that

$I=f_{x_1}<......<f_{x_5} f_{x_1}\left(x_1\right)......f_{x_5}\left(x_5\right)d{x}_1.....d{x}_5$

The integral in the most inner brackets is equal to I -F (x₄). Hence the integral in second most inner bracket is equal to

${\int }_{x_3}^{\infty }{f}_{x_4}\left(x_4\right).(1-(F\left(x_4\right)d{x}_4$

Using the substitution $u_4=1-F\left(x_4\right)$ we have that $d{u}_4=-f{x}_{4}d{x}_4$ and boundaries goes from $1-F\left(x_3\right)$ to zero. Hence,

${\int }_{x_3}^{\infty }{f}_{x_4}\left(x_4\right).(1-F(x_4\left)\right)d{x}_4={\int }_0^{1-F\left(x_3\right)}{u}_{4}d{u}_4=\frac{(1-F{\left(x_3\right))}^2}{2}$

 

Step 2: Explanation (Part b)

Observe the sequence that has been made here. Sub integration functions are as follows

$$\begin{gathered} 1-F\left(x_4\right),\frac{(1-F{\left(x_3\right))}^2}{2},\frac{(1-F{\left(x_2\right))}^3}{3!},\frac{(1-F{\left(x_1\right))}^4}{4} \\ I={\int }_{\mathrm{ℝ}}\frac{(1-F{\left(x_1\right))}^4}{4}f{x}_1{x}_{1}d{x}_1 \end{gathered}$$

and with $u_1=1-F\left(x_1\right)$ we end up with

${\int }_{\mathrm{ℝ}}\frac{(1-F{\left(x_1\right))}^4}{4}f{x}_1{x}_{1}d{x}_1={\int }_0^1\frac{{u^4}_1}{4!}d{u}_1=\frac{1}{5!}$

Step 3: Explanation (Part c)

Since variable $x1,.....x5$ are independent and equally distributed they can order themselves in any possible way with the same probability.