13. Creating Formulas
Question 1
E(Y) = E(X1 + ... + Xn) = E(X1) + ... + E(Xn) = μ + ... + μ = nμ.
Assuming the Xj are all independent,
var(Y) = var(X1) + ... + var(Xn) = = σ2 + ... + σ2 = nσ2.
Thus sd(Y) = √n σ.
Question 2
The distribution for X given by Table 11.1 is just the Binomial(3, 0.5) distribution. Using the formulas from Section 13.3 we find
E(X) = 3 × 0.5 = 1.5, and
sd(X) = √(3 × 0.5 × 0.5) = √0.75 = 0.866.
Of course these are the same as the values obtained from Table 11.1 using the original definitions of expected value and standard deviation.