1. ## independent random variables

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

(a) What is the joint cumulative distribution function of $A, B, C?$
(b) What is the probability that all the roots of the equation $Ax^2 + BX + C = 0$ are real?

2. Hello,

Let x,y,z in $\mathbb{R}$

The cdf of (A,B,C) is $P(A\leq x,B\leq y,C\leq z)=P(\{A\leq x\}\cap \{B\leq y\} \cap \{C\leq z\})$

But since A,B,C are independent, the probability of the intersection is the product of the probabilities

So the cdf is $P(A\leq x)P(B\leq y)P(C\leq z)$
And since you certainly know the cdf of a uniform distribution, you'll be able to answer the question...

You have to find $P(B^2-4AC\geq 0)=\int_0^1\int_0^1 \int_{2\sqrt{ac}}^1 db ~da ~dc$

Try to understand that...