Supose that the average distance between a random variable X and a constant c is measured by the function E (X - c)^2.
Note that
1) E means the expectation/expected value of a random variable. And,
2) E (X- c)^2 can be viewed as a function of c: Q(c) = E (X - c)^2.
Show that E(X-c)^2 = E (X - E(X))^2 + (E(X) - c)^2 ?
Note that
1) E means the expectation/expected value of a random variable. And,
2) E (X- c)^2 can be viewed as a function of c: Q(c) = E (X - c)^2.
Show that E(X-c)^2 = E (X - E(X))^2 + (E(X) - c)^2 ?
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Writing m for E(X),
E(X-c)^2=E(X-m+m-c)^2=E((X-m)^2+(m-c)^… +0 (the last term is zero because E(X-m)=0 ) so we are done
E(X-c)^2=E(X-m+m-c)^2=E((X-m)^2+(m-c)^… +0 (the last term is zero because E(X-m)=0 ) so we are done