Contents
How do you prove an inequality is convex?
Let f(x) be convex on the interval I. Suppose a 0 is a real number for which a + ϵ ≤ b − ϵ. Then f(a) + f(b) ≥ f(a + ϵ) + f(b − ϵ). If f is strictly convex, then the inequality is strict.
Is e x convex?
The function ex is differentiable, and its second derivative is ex > 0, so that it is (strictly) convex. Hence by a result in the text the set of points above its graph, {(x, y): y ≥ ex} is convex.
What is Jensen’s inequality used for?
Jensen’s Inequality is a useful tool in mathematics, specifically in applied fields such as probability and statistics. For example, it is often used as a tool in mathematical proofs. It is also used to make claims about a function where little is known or needs to be known about the distribution.
Is ex convex or concave?
The logarithm f(x) = log x is concave on the interval 0 ex is convex everywhere.
Can ex ever concave?
Determining concavity obviously requires finding the second derivative, if it even exists. Example: The graph of ex is always concave up because the second derivative of ex is ex, which is positive for all real numbers. For all other values besides the inflection points f′′(x)<0 and thus the graph is concave.
How to use Jensen’s inequality in the real world?
To use Jensen’s inequality, we need to determine if a function g is convex. A useful method is the second derivative. A twice-differentiable function g: I → R is convex if and only if g ″ ( x) ≥ 0 for all x ∈ I . For example, if g ( x) = x 2, then g ″ ( x) = 2 ≥ 0, thus g ( x) = x 2 is convex over R.
Can you prove Jensen’s inequality with limiting arguments?
So we can prove the Jensen’s inequality in this case. Using limiting arguments, this result can be extended to other types of random variables. E [ g ( X)] ≥ g ( E [ X]). To use Jensen’s inequality, we need to determine if a function g is convex.
What does Jensen’s inequality say about a convex function?
Jensen’s inequality states that, for any convex function g, we have E[g(X)] ≥ g(E[X]). So what is a convex function? Figure 6.2 depicts a convex function. A function is convex if, when you pick any two points on the graph of the function and draw a line segment between the two points, the entire segment lies above the graph.
Where does the secant line lie in Jensen’s inequality?
Jensen’s inequality generalizes the statement that the secant line of a convex function lies above the graph of the function, which is Jensen’s inequality for two points: the secant line consists of weighted means of the convex function (for t ∈ [0,1]), while the graph of the function is the convex function of the weighted means,