An alternative proof of Theorem 5.2 may be based on the fact that if X₁, X₂, . . ., and X_n are independent random variables having the same Bernoulli distribution with the parameter ., then Y = X₁ + X₂ + · · · + X_n is a random variable having the binomial distribution with the parameters n and .. 

(a) Verify directly (that is, without making use of the fact that the Bernoulli distribution is a special case of the binomial distribution) that the mean and the variance of the Bernoulli distribution are µ = θ and σ² = θ(1 – θ). 

(b) Based on Theorem 4.14 and its corollary on pages 135 and 136, show that if X₁, X₂, . . ., and X_n are independent random variables having the same Bernoulli distribution with the parameter θ and Y = X₁ + X₂ + · · · + X_n, then 

E(Y) = nθ and var(Y) = nθ(1 – θ)