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Math Help - probability integrals? T_T

  1. #1
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    probability integrals? T_T

    hello,
    I have this homework due tomorrow and have no idea what the first two problems are about... if anyone could give me a hand .. I would really appreciate it~~ I took the calculus class on solving these integrals and switching to polar coordinates before but, i can't recall anything right now



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  2. #2
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    Quote Originally Posted by phyo821 View Post
    hello,
    I have this homework due tomorrow and have no idea what the first two problems are about... if anyone could give me a hand .. I would really appreciate it~~ I took the calculus class on solving these integrals and switching to polar coordinates before but, i can't recall anything right now



    For convenience, make the substitutions x = \frac{v - \mu}{\sqrt{2} \sigma} \Rightarrow dv = \sqrt{2} \sigma dx and y = \frac{w - \mu}{\sqrt{2} \sigma} \Rightarrow dw = \sqrt{2} \sigma dy. Then your double integral becomes

     2 \sigma^2 \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} e^{-x^2} \, e^{-y^2} \, dx \, dy = 2 \sigma^2 \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} e^{-(x^2 + y^2)} \, dx \, dy .

    Now switch to polar coordinates by making the change of variable x = r \cos \theta and y = r \sin \theta. Note that dx \, dy \rightarrow r \, dr \, d\theta.

    Now read the first part of this: http://mathworld.wolfram.com/GaussianIntegral.html
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  3. #3
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    thanks for the help! does anybody know how to do the rest of the problems?
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  4. #4
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    \begin{aligned}<br />
   \sum\limits_{k=1}^{n}{k\binom nk}&=\sum\limits_{k=0}^{n-1}{(k+1)\binom n{k+1}} \\ <br />
 & =n\sum\limits_{k=0}^{n-1}{\binom{n-1}k} \\ <br />
 & =n2^{n-1}.\quad\blacksquare<br />
\end{aligned}
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