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pventurelli@gotoibr.com 04-28-2012 08:02 PM

Questions on Probability
 
I have a question on the use of the Expectation Value operation as used in the class and the book.

My (simplistic) understanding of the expectation value of a quantity is basically like taking a weighted average: for random variable X with probability distribution Pr(X) the expectation value would be either a sum (if X is discrete) or an integral (X is continuous) over all X values of X_i * Pr(X=X_i). And then I would multiply everything by a 1/N or a 1/Delta(X).

Is this more or less what is meant in the class by the E[] notation? Maybe I just need to become more comfortable using this operation, however language like that on page 61 of the book where the in sample error is defined as an averaging procedure in (implied?) contrast to the definition of E_out leaves me confused. Isn't expectation a weighted average *by definition*?

Thanks for your help!

yaser 04-28-2012 08:38 PM

Re: Questions on Probability
 
Quote:

Originally Posted by pventurelli@gotoibr.com (Post 1657)
I have a question on the use of the Expectation Value operation as used in the class and the book.

My (simplistic) understanding of the expectation value of a quantity is basically like taking a weighted average: for random variable X with probability distribution Pr(X) the expectation value would be either a sum (if X is discrete) or an integral (X is continuous) over all X values of X_i * Pr(X=X_i). And then I would multiply everything by a 1/N or a 1/Delta(X).

Is this more or less what is meant in the class by the E[] notation? Maybe I just need to become more comfortable using this operation, however language like that on page 61 of the book where the in sample error is defined as an averaging procedure in (implied?) contrast to the definition of E_out leaves me confused. Isn't expectation a weighted average *by definition*?

Thanks for your help!

The sample average can be technically posed as an expected value (by giving each example a probability 1/N). However, the correct way of thinking about E_{\rm in} is that it is a random variable (the source of randomization being which data set was generated) rather than an expected value like E_{\rm out}. However it is a random variable obtained by averaging a (hopefully) large number of sample points, and as such its value will tend to be close to the expected value E_{\rm out} per the laws of large numbers.


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