Expected value of two sums

Question

I am trying to derive the expected value of the following form:

$E[{\frac{\sum_{i=1}^{n} {x_i}{y_i}}{\sum_{i=1}^{n} {x_i}^2}}] = $

I understand that x is a constant, and that y is a random variable. I believe the trouble I am having involves some unfamiliarity with manipulating sums. What property should I try to apply to manipulate this to something manageable?

Answer 1

Recall the linearity of expectation to justify the following steps. $$E \left[\frac{\sum_i x_i y_i}{\sum_i x_i^2}\right] = \frac{E \left[\sum_i x_i y_i\right]}{\sum_i x_i^2} = \frac{\sum_i x_i E[y_i]}{\sum_i x_i^2}.$$ Use your model assumptions to compute $E[y_i]$.