Mathematics and Mathematical Statistics Lesson of the Day – Convex Functions and Jensen’s Inequality
September 10, 2014 Leave a comment
Consider a real-valued function that is continuous on the interval , where and are any 2 points in the domain of . Let
be the midpoint of and . Then, if
then is defined to be midpoint convex.
More generally, let’s consider any point within the interval . We can denote this arbitrary point as
then is defined to be convex. If
then is defined to be strictly convex.
There is a very elegant and powerful relationship about convex functions in mathematics and in mathematical statistics called Jensen’s inequality. It states that, for any random variable with a finite expected value and for any convex function ,
A function is defined to be concave if is convex. Thus, Jensen’s inequality can also be stated for concave functions. For any random variable with a finite expected value and for any concave function ,
In future Statistics Lessons of the Day, I will prove Jensen’s inequality and discuss some of its implications in mathematical statistics.