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

[zak100]

Hi,

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

http://ib.berkeley.edu/labs/slatkin/eriq/classes/guest_lect/mc_lecture_notes.pdf

 

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 May 31, 2020 by zak100

[mathematic]

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.

[Country Boy]

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.

[mathematic]

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.