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Math Help - how a continous can be added/subtracted with/by a discrete!?

  1. #1
    alv
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    Question how a continous can be added/subtracted with/by a discrete!?

    hello world! this is my first post to this forum.

    please answer to the following seemingly classical probability question, it's urgent for me:

    suppose X is a Poisson(1) random variable, and Z is a Standard Normal {N(0,1)} random variable. what is the result of the following probability:

    P(X > (16 - Z^4) / (X + Z^2)) < 3/8

    PS. by the sign ^ i mean power, so e.g. Z^4 is Z to the power of four.
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    Hi,

    What do you mean by "what is the result of the following probability"? Are you asking whether the statement is true or false? Or are you being asked to prove it?
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  3. #3
    alv
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    Exclamation yes meymathis is right...

    Quote Originally Posted by meymathis View Post
    Hi,

    What do you mean by "what is the result of the following probability"? Are you asking whether the statement is true or false? Or are you being asked to prove it?
    oops! yes you're right meymathis! i meant to prove that inequality. so sorry for this mistake...waiting for your replies
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    Sorry this took so long. It took me a while to get my thinker working. Hope this isn't too late!? Also, I proved something ever so slightly weaker than what you have here. I hope that there is a typo and it should have been \leq3/8

    Whenever I see a proof using with an inequality, the first thoughts through my head is Markov, Chebyshev, or Jensen's inequalities. Let's start with the most obvious, Markov. Are you allowed to use this one?

    <br />
\mathbf{P}\left( X>\frac{16-Z^4}{X+Z^2}\right) <br />

    <br />
=\mathbf{P}\left( X^2+XZ^2+Z^4>16\right) <br />

    Now use Markov.

    <br />
\leq \frac{\mathbf{E}\left[ X^2+XZ^2+Z^4\right] }{16}<br />

    <br />
  =\frac{\mathbf{E}\left[ X^2\right]+\mathbf{E}\left[XZ^2\right]+\mathbf{E}\left[Z^4\right] }{16}<br />

    Now by independence:

    <br />
  =\frac{\mathbf{E}\left[ X^2\right]+\mathbf{E}\left[X\right]\mathbf{E}\left[Z^2\right]+\mathbf{E}\left[Z^4\right]}{16} <br />

    X is Poisson(1) (Poisson Distribution -- from Wolfram MathWorld equationa 19 and 22)so
    \mathbf{E}\left[X\right] = 1

    \mathbf{E}\left[X^2\right] = 2

    Z is standard normal (http://en.wikipedia.org/wiki/Normal_distribution#Moments) so
    \mathbf{E}\left[Z^2\right] = 1

    \mathbf{E}\left[Z^4\right] = 3

    That get's it except that its \leq rather than <.
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  5. #5
    Junior Member frenzy's Avatar
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    Quote Originally Posted by meymathis View Post
    ...
    <br />
=\mathbf{P}\left( X^2+XZ^2+Z^4>16\right) <br />

    Now use Markov.

    <br />
\leq \frac{\mathbf{E}\left[ X^2+XZ^2+Z^4\right] }{16}<br />

    ...

    That get's it except that its \leq rather than <.
    Since you have \mathbf{P}\left( X^2+XZ^2+Z^4>16\right) (not \mathbf{P}\left( X^2+XZ^2+Z^4\geq16\right))

    you get ...

    <br />
=\mathbf{P}\left( X^2+XZ^2+Z^4>16\right)< \frac{\mathbf{E}\left[ X^2+XZ^2+Z^4\right] }{16}<br />
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    Thanks Frenzy. Details, details, details....

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  7. #7
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    Quote Originally Posted by frenzy View Post
    Since you have \mathbf{P}\left( X^2+XZ^2+Z^4>16\right) (not \mathbf{P}\left( X^2+XZ^2+Z^4\geq16\right))

    you get ...

    <br />
=\mathbf{P}\left( X^2+XZ^2+Z^4>16\right)< \frac{\mathbf{E}\left[ X^2+XZ^2+Z^4\right] }{16}<br />
    You know actually, I don't believe I've seen this before. Not that its hard, but it does not follow from Markov. The proof is almost exactly the same. Depending on the teacher, they may demand to see the proof of this variation of Markov. See the proof at http://en.wikipedia.org/wiki/Markov's_inequality#Special_case:_probability_theo ry but just adapt the X\geq a to X>a
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