Hi, I appreciate if someone could clear 2 following doubts about the screenshot.

Question

Hi,

I appreciate if someone could clear 2 following doubts about the screenshot.

What is the meaning of this statement in following screenshot(consider F and epsilon are fixed), if F(x) is fixed, does it mean I shld consider F(X) or epsilon as something like (F(x)=fixed number e.g. f(x)=6 or epsilon=.001) or it has another meaning? Besides, if we develope left side of formula, I will have E(epsilon) in the formula, if we considered epsilon as a fixed number, then what would be the emaning of Expected value for a fixed number?! As I know, expected value is considered for random variables and epsilon is not a random variable here.

Explanation / Answer

1. Short Answer: F(x) is fixed, epsilon is not fixed. And yes, if both F(x) and X are fixed, we can treat them as numbers.

Detailed answer :

Note that F is fixed but epsilon is not fixed. Infact, the expectation in this question, is taken with respect to the randomness in epsilon. Please note, that whenever we write E( something ), the first basic question to ask is "This Expectation is taken with respect to which random variable (or what is the source of randomness in this expression) ? ".

If we ask this question in the given problem, we find that the EPSILON is the random variable with respect to which the expectation is taken because as per the given information, X is fixed and F is also fixed i.e. random variable X has already been assumed to have taken some realized value and F is also fixed. Hence, the expectation is with respect to EPSILON.

2. Short Answer: As clarified in question 1, Epsilon is not a fixed number. Usually, error terms are assumed to have Zero mean. Hence, E(epsilon) = 0 and therefore that term disapperas. Expected value of a fixed number is the number itself. Think of it as a Random variable having Probability =1 of taking that fixed value and zero for any other value.

Long answer : If we expand left hand side,

E[ ( f(x) + e - f_hat(x) )^2 ] = E[ (f(x) - f_hat(x))^2 + 2*(f(x)-f_hat(x))*e + (e^2) ]

= E[ (f(x) - f_hat(x))^2 ] + 2*(f(x)-f_hat(x))*E(e) + E[(e^2)]

Now, E(e) = 0. and  E[(e^2)] = Var(e)., and

because f and f_hat are fixed, E[ (f(x) - f_hat(x))^2 ]= (f(x) - f_hat(x))^2

Therefore,  E[ ( f(x) + e - f_hat(x) )^2 ] = (f(x) - f_hat(x))^2 +  Var(e)

Dear student,If you find this answer helpful, do provide your valuable feedback through voting and commenting.

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