Understanding the Equation of Monte Carlo Method returning value to the expected function

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Hi,

I was reading an article related to MonteCarlo Method. The link of the article is:

I found the equation in the following attached image.

1)In the equation related to the attached image, we are assigning a value to a function. What does this mean?

2)What the expected function will do with this value?

3)Please guide me by providing examples of function fX(ꭓ), E(g(X)) and g(ꭓ).

Zulfi.

Edited by zak100
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The statement is a little confusing.  I believe its intent was to consider only those values of X which have a probability greater than zero.

Simple example: X has 3 values 0,1,-1. f(-1)=1/4, f(0)=1/2, and f(1)= 1/4.  g(-1)=0, g(0)=1, g(1)=2.  Then E(g(X))=1.

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• 1 year later...
On 5/31/2020 at 4:53 PM, mathematic said:

The statement is a little confusing.  I believe its intent was to consider only those values of X which have a probability greater than zero.

Simple example: X has 3 values 0,1,-1. f(-1)=1/4, f(0)=1/2, and f(1)= 1/4.  g(-1)=0, g(0)=1, g(1)=2.  Then E(g(X))=1.

You should state that the three values of X are assumed to have equal probability so P(-1)= P(0)= P(1)= 1/3.  Then E(g(x))= g(-1)/3+ g(0)/3+ g(1)/3= 0/3+ 1/3+ 2/3= 1.

Zak100, it is not the "expected function" but the "expected value" of the function.

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8 hours ago, Country Boy said:

You should state that the three values of X are assumed to have equal probability so P(-1)= P(0)= P(1)= 1/3.  Then E(g(x))= g(-1)/3+ g(0)/3+ g(1)/3= 0/3+ 1/3+ 2/3= 1.

Zak100, it is not the "expected function" but the "expected value" of the function.

The probabilities I assigned also work.  Equality is not needed.

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