b******r
发帖数: 120 | 1 Let X be a random variable. What is bigger, E(e^X) or e^(E(X))? |
l******f
发帖数: 568 | 2 Jensen inequality
【在 b******r 的大作中提到】
: Let X be a random variable. What is bigger, E(e^X) or e^(E(X))?
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b***k
发帖数: 2673 | 3 I'd like to write it out.
E[e^x]=e^E[x]*E[e^(x-E[x])]>=e^E[x]*E[1+x-E[x]]=e^E[x]
Jensen inequality
【在 l******f 的大作中提到】
: Jensen inequality
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b***k
发帖数: 2673 | 4 E[e^x]=e^E[x]*e^(-E[x])*E[e^x]=e^E[x]*E[e^(-E[x])*e^x]=e^E[x]*E[e^(x-E[x])]
中间用到 a*E[x]=E[a*x], where a=constant.
clear enough? :-) |
c*****s
发帖数: 385 | 5 定理：
if f(x) is a convex function, E[f(x)] >= f(E[x]), just draw a graph and you
will see E[f(x)] is always above f(E[x]).
【在 b******r 的大作中提到】
: Let X be a random variable. What is bigger, E(e^X) or e^(E(X))?
|
l******f
发帖数: 568 | 6 let f(x) be a convex function (e^x in this context) and l(x)=a+b*x be a
tangent line to f(x) at the point f(E(x)) for some a and b.
by convexity f(x)>=a+b*x=l(x)
Take expectation on both sides, E(f(x))>=E(a+b*x)=a+b*E(x)=l(E(x))=f(E(x))
【在 b***k 的大作中提到】
: E[e^x]=e^E[x]*e^(-E[x])*E[e^x]=e^E[x]*E[e^(-E[x])*e^x]=e^E[x]*E[e^(x-E[x])]
: 中间用到 a*E[x]=E[a*x], where a=constant.
: clear enough? :-)
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