An alternative proof of theorem 2 may be based on the fact

Question

An alternative proof of Theorem 2 may be based on the fact that if X₁, X₂, ..., and Xn are independent random variables having the same Bernoulli distribution with the parameter θ, then Y = X₁ + X₂ +···+ X_n is a random variable having the binomial distribution with the parameters n and θ. 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 ).

Theorem 2

The mean and the variance of the binomial distribution are