18.14 Suppose that Yi, i = 1, 2,

c, n, are i.i.d. with E(Yi) = m, var(Yi) = s2, and finite fourth moments. Show the following:

a. E1Y2i 2 = m2 + s2.

b. Y ¡ p m.

c.

1 na n

i = 1 Y2i

¡ p m2 + s2.

d.

1 na n

i = 1

(Yi - Y)2 =

1 na n

i = 1 Y2i

- Y2.

e.

1 na n

i = 1

(Yi - Y)2 ¡ p s2.

f. s2 =

1 n - 1a n

i = 1

(Yi - Y)2 ¡ p s2.