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Math Help - Probability of a quadratic equation to have a solution

  1. #1
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    Probability of a quadratic equation to have a solution

    Hi,

    Let ksi, eta and dzeta to be iid rv ~ Uniform(0,1). What is the probability that the equation: ksi*x^2 + eta*x + dzeta = 0, to have a real solution i.e. D>=0. I got that I need to specify the limits of the tripple integral I have to solve for finding this probability... The point is that I don't know how to solve: (eta^2)/4>=ksi*dzeta. The outer integral would be from 0 to 1, but what about the inner two?

    Some help will be much appreciated. Thanks.
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  2. #2
    MHF Contributor matheagle's Avatar
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    This is just a nasty calc problem.
    You want to integrate the density of 1, over the region

    y^2>4xz but only in the unit cube.

    The best way to start is to find the boundary, when y^2=4xz

    and then try to make sense of your region, or the complementary region.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by dely84 View Post
    Hi,

    Let ksi, eta and dzeta to be iid rv ~ Uniform(0,1). What is the probability that the equation: ksi*x^2 + eta*x + dzeta = 0, to have a real solution i.e. D>=0. I got that I need to specify the limits of the tripple integral I have to solve for finding this probability... The point is that I don't know how to solve: (eta^2)/4>=ksi*dzeta. The outer integral would be from 0 to 1, but what about the inner two?

    Some help will be much appreciated. Thanks.
    This is quite a nice problem for demonstrating hit-and-miss Monte-Carlo. We generate a large sample of points in the unit cube count number of cases where discriminant is positive and divide by the sample size.

    Code:
    >NN=10000000;       ..sample size
    >
    >xx=random(3,NN);   ..sample of size NN of coefficients
    >
    >pp=sum(xx(1,: )^2>4*xx(2,: )*xx(3,: ))/NN ..count the positive disc and divide by sample size
         0.254644 
    >
    >
    >SE=sqrt((NN*pp*(1-pp)))/NN  ..approx standard error of the estimate
         0.000137768 
    >
    So the required probability is ~=0.255 or 25.5%

    CB
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  4. #4
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    Quote Originally Posted by matheagle View Post
    This is just a nasty calc problem.
    You want to integrate the density of 1, over the region

    y^2>4xz but only in the unit cube.

    The best way to start is to find the boundary, when y^2=4xz

    and then try to make sense of your region, or the complementary region.
    Hi,
    Yeah, that is exactly the issue. I am not sure how to do that. Could you please elaborate on how to define the boundaries of the integration over the triple integral. I could solve it just if new how to setup the last boundary.
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  5. #5
    Guy
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    There's actually another route you can go with this. You can make use of the following fact:

    If X ~ uniform[0, 1] then -log(X) ~ exponential(1).

    After an annoying amount of calculus and some standard results from distribution theory, the answer I get is:

    \displaystyle<br />
\frac{3\log 4 + 5}{36} \approx 0.2544<br />
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  6. #6
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    Quote Originally Posted by Guy View Post
    There's actually another route you can go with this. You can make use of the following fact:

    If X ~ uniform[0, 1] then -log(X) ~ exponential(1).

    After an annoying amount of calculus and some standard results from distribution theory, the answer I get is:

    \displaystyle<br />
\frac{3\log 4 + 5}{36} \approx 0.2544<br />
    Hi, Thanks for the answer. Could you please show the steps for this transformation. I need to see how to form the integral. If you could post it and I will sove it for me.
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  7. #7
    Guy
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    Quote Originally Posted by dely84 View Post
    Hi, Thanks for the answer. Could you please show the steps for this transformation. I need to see how to form the integral. If you could post it and I will sove it for me.
    Well, if x, y, z ~ iid uniform[0, 1] then

    \displaystyle<br />
P(y^2 \ge 4xy) = P(-2\log y \le - \log x - \log y - \log 4).<br />

    Now, -2log(y) ~ exponential(2) [I'm going to use the scale parametrizations of the exponential and gamma], and because x and z are independent, -log(x) - log(z) is the sum of two independent exponential(1) random variables, and so is distributed gamma(2, 1). So, if A ~ exponential(2) and B ~ gamma(2, 1) with A independent of B you want to calculate

    <br />
P(A \le B - \log(4)) = P(A \le B - \log(4) \cap B \ge \log(4)).<br />

    It should be obvious now how to proceed. If you don't know from class the result about -log(x) giving an exponential, it is really very easy to show using the so-called distribution function method:

    <br />
P(-\log (X) \le x) = P(X \ge e^{-x}) = 1 - e^{-x}<br />

    which is the cdf of an exponential random variable.
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