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Find the probability that $x^2 - 2ax + b$ has complex roots if the coefficients $a$ and $b$ are independent random variables with the common density

  1. uniform, that is $1/h$, and
  2. exponential, that is $\alpha e^{-\alpha x}$

This comes down to finding $P(a^2 \lt b)$. But since $a$ and $b$ are both random variables, would it be $P(a^2\lt b) = P(x\lt k)P(y \lt k^2)$? That doesn't seem particularly correct.

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    I've tried to fix your TeX. Please check if I have introduced any mistakes. Moreover, you probably mean non-real roots, otherwise it would be trivially $1$ :)2011-04-18
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    For 1, is the range of $a,b \ [0,h] \text{ or } [\frac{-h}{2},\frac{h}{2}]$ or what?2011-04-18
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    I guess $a$ and $b$ are independent?2011-04-18
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    the range of 1) is 00. And yes, a and b are uncorrelated2011-04-18
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    The $a$ in the quadratic is probably not the same $a$ in $ae^{-ax}$ so you should change the latter $a$ to something else. As it is, sampling $a$ from the density $ae^{-ax}$ doesn't make much sense in this context.2011-04-19
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    Nitpick: The probability is 1. The polynomial always has complex roots, as the reals are a subset of the complex numbers. Perhaps the question should be "non-real" roots...2011-04-19

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I have seen several similar questions here. The idea is to use the joint density function of $a$ and $b$, which is (assuming independence), (1) $f(x,y)=\frac1{h^2}$ in the square $[0,h]^2$ and 0 otherwise; (2) $f(x,y)=\lambda^2e^{-\lambda(x+y)}$ in the first quadrant and 0 otherwise (I replace the parameter $a$ by $\lambda$ because it's confusing with a r.v. $a$.)

In both cases: $$P(a^21$ should be similar. $$P(a^2

Case (2) can be solved similarly with a different integral, and I'll leave it to you.

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    I recently was going through the same problem. I proceeded in a similar way. However the answer given was h/3 for 02017-03-19
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I'm guessing that you need to add "uncorrelated" to the description of the random variables a and b. In that case, you will have to integrate over the possible values. For the uniform case this is easy, the probability is proportional to the area. For the other, I'm going to let you think about it and see what you come up with.

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    My idea for the exponential is to do a double integral, with paramters -b^1/2 to b^1/2 and parameters for b from 0 to infinity. Would I then just plug in the probability density functions for a and b as the integrand? Thanks so much!2011-04-18
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    @zamenek; And I guess you realize that you only want to integrate over the area where the result is whatever it is you want (i.e. complex).2011-04-18
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    And I guess that by *uncorrelated* what you really mean is *independent*.2011-04-19
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    @zamenek Did you notice that the beginning of @GWu's post applies to *any* distribution of $(a,b)$? This answers your question to @Carl.2011-04-19
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    @Didier; Yes! That's a problem with having taken the class back in the 1970s.2011-04-19
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The polynomial has complex roots if and only if (2a)^2 - 4b <0. That means $a^2 -b <0.$ Now all you have to calculate $P(A^2 < B)$ with both of those distributions.

That I will leave to you (or another poster) to do.