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Math Help - Integrate PDF

  1. #1
    Member courteous's Avatar
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    Question Integrate PDF

    Random variable (X,Y) has PDF f(x,y)=\frac{C}{\pi^2(x^2+20)(y^2+45)}. What is C?
    I set up the equation, but don't know the integration part.
    \displaystyle{ \int_{-\infty}^\infty\int_{-\infty}^\infty f(x,y) dy dx = 1 = \frac{1}{\pi^2}\int_{-\infty}^\infty \int_{-\infty}^\infty \frac{C}{(x^2+20)(y^2+45)} dy dx }

    C should be 30.
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  2. #2
    Flow Master
    mr fantastic's Avatar
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    Quote Originally Posted by courteous View Post
    I set up the equation, but don't know the integration part.
    \displaystyle{ \int_{-\infty}^\infty\int_{-\infty}^\infty f(x,y) dy dx = 1 = \frac{1}{\pi^2}\int_{-\infty}^\infty \int_{-\infty}^\infty \frac{C}{(x^2+20)(y^2+45)} dy dx }

    C should be 30.
    The double integral is seperable using a simple corrollary to Fubini's stronger theorem:

    \displaystyle \int_{-\infty}^\infty \frac{1}{x^2+20} \, dx \int_{-\infty}^\infty \frac{1}{y^2+45} \, dy.

    Each integral involves the inverse tan function.
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  3. #3
    Member courteous's Avatar
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    \displaystyle{  \int_{-\infty}^{\infty}\frac{1}{y^2+45}dy\overbrace{=}^{s  ub} \int_{\theta=-\frac{\pi}{2}}^{\theta=\frac{\pi}{2}}\frac{\sqrt{4  5}\frac{1}{cos^2(\theta)}}{45(\tan^2(\theta)+1)}d\  theta = \frac{1}{\sqrt{45}}\left[\theta\right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}}=\frac{\pi}{\sqrt{45  }} }

    Similar result for x: \displaystyle{  \frac{\pi}{\sqrt{20}}.

    Which finally gives \displaystyle{  1=\frac{C}{\pi^2}\frac{\pi}{\sqrt{20}}\frac{\pi}{\  sqrt{45}}=\frac{C}{30} \Rightarrow C=30  } A piece of ... I'm so ignorant.

    Why does it work? Wikipedia doesn't say much (to me) under strong versions.
    I'm really asking whether there is a natural learning path that step-by-step makes the whole picture clear (seeing forest instead of trees)?

    Let's take calculus. Would this be anywhere near a step-by-step "learning signpost"?
    1. Calculus (C)
    2. Multivariable C
    3. Vector C
    4. Analytic geometry
    5. Topology
    6. ?

    Do these various theorems ever start "falling in place"?
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