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Random Variable Function Inequality For Real Numbers Interval Arithmetic Mean Weight Lie Algebras Expected Value Range Convex Function

1	Random Variable
1	Inequality For Real Numbers
1	Interval
1	Function
1	Arithmetic Mean
2	Weight Lie Algebras
3	Expected Value
4	Convex Function
6	Range

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Jensens Inequality

If is a convex function on the interval , for each $\left\{x_k\right\}_{k=1}^n \in[a,b]$ and each $\left\{\mu_k\right\}_{k=1}^n$ with $\mu_{k}\geq0$ one has:

$\begin{displaymath}f\left(\frac{\sum_{k=1}^{n}\mu_{k}x_{k}}{\sum_{k}^{n}\mu_{k}}... ...\sum_{k=1}^{n}\mu_{k}f\left(x_{k}\right)}{\sum_{k}^{n}\mu_{k}}.\end{displaymath}$

A common situation occurs when $\mu_1+\mu_2+\cdots+\mu_n=1$ ; in this case, the inequality simplifies to:

$\begin{displaymath}f\left(\sum_{k=1}^n \mu_k x_k\right)\leq \sum_{k=1}^n \mu_k f(x_k)\end{displaymath}$

where $0\le \mu_k\le 1$ .

If is a concave function, the inequality is reversed.

Example:
is a convex function on . Then

$\begin{displaymath}(0.2\cdot4+ 0.5\cdot3+0.3\cdot7)^2 \leq 0.2(4^2) + 0.5(3^2)+0.3(7^2).\end{displaymath}$

A very special case of this inequality is when $\mu_k=\frac{1}{n}$ because then

$\begin{displaymath}f\left(\frac{1}{n}\sum_{k=1}^n x_k\right)\le\frac{1}{n}\sum_{k=1}^n f(x_k)\end{displaymath}$

that is, the value of the function at the mean of the

is less or equal than the mean of the values of the function at each

There is another formulation of Jensen's inequality used in probability:
Let be some random variable, and let be a convex function (defined at least on a segment containing the range of ). Then the expected value of is at least the value of at the mean of :

$\begin{displaymath} \mathrm{E}[f(X)] \ge f(\mathrm{E}[ X]). \end{displaymath}$

With this approach, the weights of the first form can be seen as probabilities.