Show that EX c 2 is minimized by taking c EXSolutionConside

Show that E(X c)^ 2 is minimized by taking c = E(X).

Solution

Consider

E(X-c)²

= E(x²-2x c+c²) ( by expansing the square

Use lienar property of Expectation

= E(X²)-2cE(X)+E(C²)

= E(X²)-2cE(X)+n(C²)

Let f(c) = E(X²)-2cE(X)+(C²)

To find maximum or minimum use first derivative =0 and second der = positive

f\'(c) = 0-2E(X)+2c

f\"(C) = 0+2>0

Equate f\'(c) =0 to get E(x) = c

Hence f(c) i.e.

E(X c)^ 2 is minimized by taking c = E(X).