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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 +···+ Xn 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
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