All $a_i$,$b_i$ and $c_i$ $(1<i<10)$ are independent standard normal distributed variable by each other ($a_i$,$b_i$,$c_i$$∼$$N(0,1)$).
and I would like to know about expected value of combination of these variables, which is $$E\biggr[\sum_{i=0}^{10} (\frac{a^2c^2}{a^2+b^2})\biggr]$$
I think denominator follows Chi-square distribution
but this equation seems that denominator and numerator are not independent so I have trouble solving this.
what is expected value of above equation?
You can do this without any heavy computation. First note that $c_i^{2}$ is independent of $a_i$'s and $b_i$'s. Next note that $E\sum \frac {a_i^{2}} {a_i^{2}+b_i^{2}} =E\sum \frac {b_i^{2}} {a_i^{2}+b_i^{2}} $ (in view of the assumptions on $a_i$'s and $b_i$'s) and this two add up to $10$). Hence the answer is $5$.