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Chapter 7: Q.7.1 (page 359)

Show that E[(X鈭�a)2] is minimized at a=E[X].

Short Answer

Expert verified

DifferentiateE[(X鈭�a)2]respective toa.

Step by step solution

01

Given Information

E[(X鈭�a)2]is minimizing ata=E[X].

02

Explanation

We have that

E(X-a)2=EX2+a2-2Xa

=EX2+a2-2aE(X)

Using the differentiation respective to a and equalizing it to zero, we have that

ddaE(X-a)2=2a-2E(X)

=0

03

Explanation

which yields that a=E(X).

So, E(X-a)2is minimized whena=E(X) which had to be proved.

04

Final Answer

E(X-a)2 is minimized whena=E(X) which had to be proved.

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