Ratio of sum of squares of Normal Distributions Find $$E[\frac{\sum_{i=1}^{51} X_i^2}{\sum_{i=51}^{101} X_i^2}], $$if $X_1,X_2,...,X_{51}$ are independent and are distributed as N(0,1). This question appeared in my final exam of probability and stochastic processes, that occurred today .I couldn't answer this in the exam and would love to find out more. I did simplify this a bit though. Here's what I did:
$Y = \sum_{i=1}^{50} X_i^2$, $Z = \sum_{i=52}^{101} X_i^2,X = X_{51}$. Then, 
$X \sim \gamma(1/2,1/2), Y \sim \gamma(25,1/2), Z \sim\gamma(25,1/2). $ Then we wish to find $E[\frac{X+Y}{X+Z}]$, where X,Y,Z are independent random variables. I was stuck after this, as there was this integral of $$\frac{x^{-1/2}}{x+z}e^{-25x}, 0\leq x < \infty$$ Any help will be appreciated. 

You can write the ratio as $\frac{Y}{X+Z} + \frac{X}{X+Z}$. Then $\frac{Y}{X+Z}$ has an F-distribution with $d_1 = 50$ and $d_2 = 51$ degrees of freedom. And $\frac{X}{X+Z}$ has a Beta-distribution with parameters $\alpha = \frac{1}{2}, \beta = \frac{50}{2} = 25$ .
