Jensens's inequality concerns the expected value of convex and concave transformations of a random variable.
The following is a formal statement of the inequality.
Proposition Let be an integrable random variable. Let be a convex function such thatis also integrable. Then, the following inequality, called Jensen's inequality, holds:
A function is convex if, for any point the graph of lies entirely above its tangent at the point :where is the slope of the tangent. Setting and , the inequality becomesTaking the expected value of both sides of the inequality and using the fact that the expected value operator preserves inequalities, we obtain
If the function is strictly convex and is not almost surely constant, then we have a strict inequality:
A function is strictly convex if, for any point the graph of lies entirely above its tangent at the point (and stricly so for points different from ):where is the slope of the tangent. Setting and , the inequality becomesand, of course, when . Taking the expected value of both sides of the inequality and using the fact that the expected value operator preserves inequalities, we obtainwhere the first inequality is strict because we have assumed that is not almost surely constant and therefore the eventdoes not have probability .
If the function is concave, then
If is concave, then is convex and by Jensen's inequality:Multiplying both sides by and using the linearity of the expected value we obtain the result.
If the function is strictly concave and is not almost surely constant, then
Similar to previous proof.
Suppose a strictly positive random variable has expected valueand it is not constant with probability one. What can we say about the expected value of , by using Jensen's inequality?
The natural logarithm is a strictly concave function, because its second derivative is strictly negative on its domain of definition.
As a consequence, by Jensen's inequality, we have
Therefore, has a strictly negative expected value.
Let be a strictly positive random variable, such thatWhat can you infer, using Jensen's inequality, about the following expected value:
The functionhas first derivativeand second derivativeThe second derivative is strictly negative on the domain of definition of the function. Therefore, the function is strictly concave. Furthermore, is not almost surely constant, because it has strictly positive variance. Hence, by Jensen's inequality:
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