Solve Jensen's Inequality: Proof & Troubleshooting

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The discussion revolves around the proof of Jensen's Inequality, specifically addressing difficulties in transitioning from the expectation of a convex function to the inequality's conclusion. The theorem states that for a random variable X with a finite expectation and a convex function f, the inequality f(E(X)) ≤ E(f(X)) holds. The user struggles with the step from E(f(X)) to f(E(X)), particularly in handling the expectation of the linear combination involving λ. After some analysis, the user realizes the mistake in incorrectly factoring out λ, leading to a clearer understanding of the proof. The conversation highlights common challenges in grasping concepts of probability and expectations in mathematical proofs.
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I am having trouble with the following proof of Jensen's Inequality. I'll post the statement of the theorem, it's proof, and where I'm having problems:

Let X be a random variable
with E(X) < \infty, and let f : \mathbb{R}\rightarrow\mathbb{R} be a convex function. Then
\begin{equation*}f(E(X))\leq E(f(X))\end{equation*} [1]​

where a function is convex if \forall x_0\in \mathbb{R},\ \exists \lambda \in \mathbb{R}: f(x)\geq \lambda(x-x_0)+f(x_0)

Proof: Let f be convex and let \lambda \in \mathbb{R} be such that
f(X)\geq \lambda(x-E(X))+f(E(X)) [2]​
then
E(f(x))\geq E(\lambda(x-E(X))+f(E(X))) [3]
=f(E(X)) [4]​
Q.E.D




As is probably clear from my having problems with this, probability and dealing with expectations isn't my strong point but getting from [3] to [4] isn't looking obvious to me at all since if I expand RHS of [3] (and assume x is meant to be X, a typo) then unless I'm wrong I get:

E(f(X))\geq E(\lambda(X-E(X))+f(E(X)))=E(\lambda(X-E(X)))+E(f(E(X))) (using E(g(X)+h(X))=E(h(X))+E(f(X)))
=\lambda E(X)-E(X)+f(E(X)) (using E(aX+b)=aE(X)+b and E(a)=a (where a,b are constants))
=(\lambda-1)E(X)+f(E(X))
and this isn't [4] :confused:
Where am I going wrong?
 
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E is a linear function. Therefore E(λ(x-E(x))=λ(E(x-E(x))=λ(E(x)-E(x))=0.
 
aggh...Now I see where I've pulled the lambda out incorrectly it's obvious! cheers :)
 
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