A common situation occurs when ; in this case, the inequality simplifies to:
If is a concave function, the inequality is reversed.
is a convex function on . Then
A very special case of this inequality is when because then
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 :
With this approach, the weights of the first form can be seen as probabilities.
|Date of creation||2013-03-22 11:46:30|
|Last modified on||2013-03-22 11:46:30|
|Owner||Andrea Ambrosio (7332)|
|Last modified by||Andrea Ambrosio (7332)|
|Author||Andrea Ambrosio (7332)|