An alternative proof of Theorem 5.2 may be based on the fact that if X 1 ,

Question:

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)

Distribution
The word "distribution" has several meanings in the financial world, most of them pertaining to the payment of assets from a fund, account, or individual security to an investor or beneficiary. Retirement account distributions are among the most...