Solve Jensen's Inequality: Proof & Troubleshooting

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In summary, the proof of Jensen's Inequality states that for a random variable X with finite expected value and a convex function f, the expected value of f(X) is greater than or equal to f(E(X)). The proof relies on the fact that the expected value function E is linear, leading to the final result of f(E(X)) in step [4].
  • #1
GregA
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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 [itex]X[/itex] be a random variable
with [itex]E(X) < \infty [/itex], and let [itex]f : \mathbb{R}\rightarrow\mathbb{R}[/itex] be a convex function. Then
[itex]\begin{equation*}f(E(X))\leq E(f(X))\end{equation*}[/itex] [1]​

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

Proof: Let [itex]f[/itex] be convex and let [itex]\lambda \in \mathbb{R}[/itex] be such that
[itex] f(X)\geq \lambda(x-E(X))+f(E(X))[/itex] [2]​
then
[itex] E(f(x))\geq E(\lambda(x-E(X))+f(E(X))) [/itex] [3]
[itex]=f(E(X))[/itex] [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 [itex]x[/itex] is meant to be [itex]X[/itex], a typo) then unless I'm wrong I get:

[itex]E(f(X))\geq E(\lambda(X-E(X))+f(E(X)))=E(\lambda(X-E(X)))+E(f(E(X)))[/itex] (using E(g(X)+h(X))=E(h(X))+E(f(X)))
[itex]=\lambda E(X)-E(X)+f(E(X))[/itex] (using E(aX+b)=aE(X)+b and E(a)=a (where a,b are constants))
[itex]=(\lambda-1)E(X)+f(E(X))[/itex]
and this isn't [4] :confused:
Where am I going wrong?
 
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  • #2
E is a linear function. Therefore E(λ(x-E(x))=λ(E(x-E(x))=λ(E(x)-E(x))=0.
 
  • #3
aggh...Now I see where I've pulled the lambda out incorrectly it's obvious! cheers :)
 

FAQ: Solve Jensen's Inequality: Proof & Troubleshooting

What is Jensen's Inequality?

Jensen's Inequality is a mathematical theorem that describes the relationship between the expected value of a convex function and the convex function of the expected value of a random variable.

What is the proof of Jensen's Inequality?

The proof of Jensen's Inequality involves using the convexity of a function to show that the expected value of the function is greater than or equal to the function of the expected value of a random variable.

How can Jensen's Inequality be used in scientific research?

Jensen's Inequality is commonly used in fields such as statistics and economics to establish important results in probability theory and optimization. It can also be applied in various scientific disciplines to prove mathematical relationships between variables.

What are some common troubleshooting techniques for Jensen's Inequality?

One common troubleshooting technique is to check the validity of the convexity assumption, as Jensen's Inequality only holds for convex functions. It is also important to ensure that the expected value and function being evaluated are defined for the same set of values.

Are there any limitations to Jensen's Inequality?

Yes, Jensen's Inequality only applies to convex functions and may not hold for non-convex functions. Additionally, it may not hold for situations where the expected value and function being evaluated are defined for different sets of values. Careful consideration and analysis of the problem is necessary to determine the applicability of Jensen's Inequality.

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