152. Afunctiong(x) isconvex if the chord connectingany two points on the function’s graph lies above the graph. When g(x) is differentiable, an equivalent condition is that for every x, the tangent line at x lies entirely on or below the graph. (See the figures below.)

Howdoesg(m)g[E(X)]comparetoE[g(X)]?

[Hint:Theequation of the tangent line at xmis y

g(m)g(m) #(xm). Use the condition of convexity, substituteXforx,andtakeexpectedvalues.Note:

Unless g(x) is linear, the resulting inequality (usually called Jensen’s inequality) is strict (rather than

); it is valid for both continuous and discrete rv’s.]