Let B and C be independent random variables both having the pdf: f(x) = cx^2 if 1

Question

Let B and C be independent random variables both having the pdf: f(x) = cx^2 if 1<x<3 and 0 otherwise, where c is a suitable positive constant. 
Calculate the probability that the quadratic equation X^2+BX+C = 0 has two real roots.

freckles · Answer

Sounds like we need to find c first

tiffany_rhodes · Answer

yeah, that's what I thought but I'm not quite sure how to go about doing that.

freckles · Answer

i think you need to find c such that
We need $$\int\limits_{1}^{3}cx^2 dx=1$$

tiffany_rhodes · Answer

c = 3/26?

freckles · Answer

yes sounds good

freckles · Answer

Thinking about that last question....
Thoughts pending...

freckles · Answer

So I know that quadratics have two real roots if
$$B^2-4AC>0$$

freckles · Answer

And A=1 here so we have

tiffany_rhodes · Answer

lol alright. Oh and why did we set the integral equal to 1?

freckles · Answer

$$B^2-4C>0$$

freckles · Answer

just like in a discrete pdf the sum of the probabilities must add up to be 1

tiffany_rhodes · Answer

oh, okay. That makes sense.

freckles · Answer

@SithsAndGiggles if you want you can look at this last question she has 
My thoughts are pending slow on it

tiffany_rhodes · Answer

Alright. I don't know if it will help but my hw says the answer is .12

anonymous · Answer

$$P(B^2-4C>0)=P(B^2>4C)=P(|B|>2\sqrt C)$$
At least two nice ways to find this sort of probability: method of distribution functions, or method of transformations. Do either of these sound familiar? (Method of moment-generating functions might also work, but that sounds brutal.)

tiffany_rhodes · Answer

The first two aren't familiar whatsoever and the moment-generating functions was discussed very briefly in my lecture.

tiffany_rhodes · Answer

were*

freckles · Answer

https://engineering.purdue.edu/~ipollak/ee302/SPRING04/exams/ex2_sol.pdf This pdf has a similar problem see problem 5

anonymous · Answer

Try finding the distribution of the rv function $B^2-4C$ using the method of distribution functions.

tiffany_rhodes · Answer

we haven't covered that.

anonymous · Answer

Sorry, I meant the MGF method, not CDF >.<

anonymous · Answer

Hmm that method might not be very useful here... What else do you know to do with functions of random variables?

tiffany_rhodes · Answer

I have to go to class and I'll ask my TA. Thank you both for trying.

anonymous · Answer

For what it's worth, you're welcome. There might be some trick to it we're not seeing. Notice that $\dfrac{3}{26}\approx.12$, that might be a clue.