Showing the simple linear OLS estimators are unbiased

Show that the simple linear regression estimators are unbiased. See comments for more details!

In response to the comments about "pulling out the constants": x1, x2, ..., xn are assumed to be observed, non-random values. Thus, xbar = (x1 + ... + xn)/n is a constant. Since each xi is a known value and xbar is a known value, each (xi - xbar) is a constant (though the constant changes for each i). Similarly, the sum the of (xi - xbar)*xi over all n will also be a known constant. Let d = sum (xi - xbar)*xi over all n. At one point, we are trying to find E(1/d * sum [(xi - xbar)*yi]). 1/d is a constant, so we get 1/d * E(sum [(x_i - xbar)*yi]). The expected value of a sum of random variables is the sum of the expected values of the random variables. So we get 1/d * (sum E[(xi - xbar)*yi]). Since each (xi - xbar) is a constant (that is potentially different for each i), we can pull that out of the expectation. So we get 1/d * (sum [(xi - xbar)*E(yi)]). The video takes care of the rest.