# 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

where .

Then, if

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.

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