6.2.7 is incorrect?

[soid]

I believe 6.2.7 solution is incorrect. It assumes V(X1+X2) = V(X1) + V(X2) which I believe is wrong.

I think the correct solution is:

X - number of heads turn up.
Xi = 1 if head turns up, 0 otherwise on i-th turn.
E[Xi] = 0 * 1/2 + 1 * 1/2 = 1/2
E[X] = E[X1] +E[X2] + E[X3] = 3/2 (also related Theorem 6.3)

E[X^2] = E[(X1+X2+X3)^2] = E[X1^2 + 2 X1 X2 + X2^2 + 2 X1 X3 + 2 X2 X3 + X3^2] =
= E[X1^2] + 2 * E[X1] * E[X2] + E[X2^2] + 2 * E[X1] * E[X3] + 2 * E[X2] * E[X3] + E[X3^2] =
= 1/2 + 2 * 1/2 * 1/2 + 1/2 + 2 * 1/2 * 1/2 + 2 * 1/2 * 1/2 + 1/2 = 3.

V[X] = E[X^2] - E[X]^2 = 3 - (3/2)^2 = 3/4
D[X] = sqrt(3)/2