Q. 6.26 - A First Course in Probability

Question

Suppose that A, B, C, are independent random variables, each being uniformly distributed over$\left(0,1\right)$.

(a) What is the joint cumulative distribution function of A, B, C? 

(b) What is the probability that all of the roots of the equation $A{X}^2+Bx+C=0$ are real? 

Step-by-Step Solution

Answer

a. Joint cumulative distribution function:

$F_{A,B,C}(a,b,c)=abc$

b. The probability that all the roots of the equation are real roots: $P(B^2\ge 4D)=\frac{\log \left(2\right)}{6}+\frac{5}{36}$

Step 1: Content Introduction

We are given, $A{x}^2+Bx+C=0$

where A, B, C are independent random variables and each being distributed over [0,1]. 

Step 2: Explanation (part a)

For $(a,b,c)\in {(0,1)}^3$

we have 

$F_{A,B,C}(a,b,c)=P\left(A\le a,B\le b,C\le c\right)=P\left(A\le a\right)P(B\le b)P(C\le c)$

where the last equality holds because the variables are independent.

we know that

A, B, C are being distributed over $[0,1]$

then we have $P(A\le a)=a$

and $P(B\le b)=b$

and $P(C\le c)=c$

Therefore, $F_{A,B,C}(a,b,c)=abc$

Step 3: Explanation (part b)

The discriminant of the equation is greater or equal to zero 

such that

$B^2-4AC\ge 0$

$B^2\ge 4AC$

Now, fid the distribution of the random variable,

$D=AC$

we have

$D\in (0,1)$

and 

for $d\in (0,1)$,

$$\begin{gathered} F_D\left(d\right)=P(D\le d)=P\left(AC\le d\right)=\int {\int }_{ac\le d}dadc \\ ={\int }_0^d{\int }_0^{1}dcda+{\int }_d^1{\int }_0^{\raisebox{1ex}{$d$}\!\left/ \!\raisebox{-1ex}{$a$}\right.}dcda \\ =d+d{\int }_d^1\frac{1}{a}da \\ =d-d\log \left(d\right) \end{gathered}$$

By differentiation:

We have 

$f_D\left(d\right)=\frac{\partial }{\partial d}{F}_D\left(d\right)=-\log \left(d\right)$

Thus the required probability,

$P(B^2\ge 4D)=\frac{\log \left(2\right)}{6}+\frac{5}{36}$

Where, in order to calculate the second integral,

we have used

$\int s^2\log \left(s\right)ds=\frac{s^3\log \left(s\right)}{3}-\frac{s^3}{9}$