1. ## 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

2. Originally Posted by phyo821
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

3. thanks for the help! does anybody know how to do the rest of the problems?

4. \begin{aligned}
\sum\limits_{k=1}^{n}{k\binom nk}&=\sum\limits_{k=0}^{n-1}{(k+1)\binom n{k+1}} \\
& =n\sum\limits_{k=0}^{n-1}{\binom{n-1}k} \\