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Found on
http://en.wikipedia.org/wiki/Jensen`s_Inequality
with equality when {math|φ(X)} is not strictly convex, e.g. when it is a straight line, or when {mvar|X} follows a degenerate distribution (i.e. is a constant). The proofs below formalize this intuitive notion. ===Proof 1 (finite form)=== ==Applications and special cases== ===Form involving a probability density function=== In probabilistic langu...
Found on
http://en.wikipedia.org/wiki/Jensen`s_inequality
Let f be convex on X, and let x be a random variable with expected value, E(x). Then, E(f(x)) >= f(E(x)). For example, let X=R and f(x) = x². Suppose x is normally distributed with mean zero and standard deviation = s. Then, E(f(x)) = E(x²) = s² >= 0 = E(x) = f(E(x)).
Found on
http://glossary.computing.society.informs.org/index.php?page=J.html
If X is a real-valued random variable with E(-X-) finite and the function g() is convex, then E[g(X)] >= g(E[X]). One application: By Jensen's inequality, E[X
2] >= (E[X])
2. Since the difference between these is the variance, we have just shown that any random variable for which E[X
2<...
Found on http://www.econterms.com/glossary.cgi?query=Jensen's+inequality
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