# Question: An alternative proof of Theorem 5 2 may be based on

An alternative proof of Theorem 5.2 may be based on the fact that if X1, X2, . . ., and Xn are independent random variables having the same Bernoulli distribution with the parameter ., then Y = X1 + X2 + · · · + Xn 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 Xn are independent random variables having the same Bernoulli distribution with the parameter θ and Y = X1 + X2 + · · · + Xn, then

E(Y) = nθ and var(Y) = nθ(1 – θ)

(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 Xn are independent random variables having the same Bernoulli distribution with the parameter θ and Y = X1 + X2 + · · · + Xn, then

E(Y) = nθ and var(Y) = nθ(1 – θ)

## Answer to relevant Questions

An expert sharpshooter misses a target 5 percent of the time. Find the probability that she will miss the target for the second time on the fifteenth shot using (a) The formula for the negative binomial distribution; (b) ...Find the mean and the variance of the hypergeometric distribution with n = 3, N = 16, and M = 10, using (a) The results of Exercise 5.64; (b) The formulas of Theorem 5.7. Check in each case whether the values of n and θ satisfy the rule of thumb for a good approximation, an excellent approximation, or neither when we want to use the Poisson distribution to approximate binomial probabilities. ...In a certain desert region the number of persons who become seriously ill each year from eating a certain poisonous plant is a random variable having a Poisson distribution with λ = 5.2. Use Table II to find the ...Use the recursion formula of Exercise 5.8 to show that for θ = 12 the binomial distribution has (a) A maximum at x = n/2 when n is even; (b) Maxima at x = n – 1 / 2 and x = n + 1 / 2 when n is odd. In exercisePost your question