13. Creating Formulas

Question 1

E(Y) = E(X₁ + ... + X_n) = E(X₁) + ... + E(X_n) = μ + ... + μ = nμ.

Assuming the X_j are all independent,

var(Y) = var(X₁) + ... + var(X_n) = = 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.