Cumulative distribution function problem question?

  • #1
Hi, so this is the example I have:

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 of the roots of the equation Ax^2 + Bx +C=0 are real?

Okay so for a), I think I get this since they are independent and are all uniformly distributed, their probability density function is still just 1. Then to get get the cumulative distribution function I just integrate this from 0 to a, 0 to b, 0 to c, which gives me just F(A,B,C)=abc which makes sense to me (I think).

I'm just lost as to what to do with b), I guess the point to have the roots be real, is it just that I have to have b^2 - 4ac be positive? So that the square roots of this can be found? This is a question in a textbook and so I still tried something which I'm sure is very wrong. I tried a triple integral having my borders be integrating from (-2sqrt(ac) to 2sqrt(ac))db (because you want b^2-4ac to be greater than or equal to 0). Then just integrating from 0 to 1 according to a and 0 to 1 according to b. If anyone has tips, solutions to help me out it would be greatly appreciated as I get a probability that is greater than 1 -_- . Thank you very much! :)
 

Answers and Replies

  • #2
mathman
Science Advisor
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[itex] \int\int(\int_{\sqrt{4ac}}^1 db)dadc=\int\int(1-2\ sqrt(ac))dadc[/itex]. The limits for a and c are a little messy. You need both non negative while ac < 1/4.
 

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