Tl;dr the "weighted average" stuff is supposed to motivate Jensen's inequality from probability. Read "f(E[X])" as "a function of weights" and "E[f(X)]" as "the weighted average of functions." If we apply the probability weights (t,1-t) to the interval endpoints a and b, we get ta+(1-t)b = W1, and if we apply the same weights to corresponding maps of the endpoints, f(a) and f(b), we get tf(a)+(1-t)f(b) = W2. The function f is concave over [a,b] if, for all weights (t,1-t), f(W1) ≥ W2. In other words, the value of a function of the weights (LHS) vs. the value of a weighted function (RHS). Jensen's inequality states that if f is *convex* (so f(W1) ≤ W2), and X is a random variable, then f(E[X]) ≤ E[f(X)]. If we think of taking the expectation of X as applying weights to the values that X can take on, then obtaining the expected value E[X] is much like getting W1. Thus, f(E[X]) is analogous to f(W1). Likewise, if we think of E[f(X)] as the weighted average of the random variable Y=f(X), then E[f(X)] weights f(X) to obtain W2.