Prove that, given the conditions of the preceding problem, \(D \mu\) reaches a maximum for the given value of \(a=\frac{1}{n} \sum_{1}^{n} p_{i}\) provided

\[ p_{1}=p_{2}=\ldots=p_{n}=a \]

Preceding Problem:

The probability that event \(A\) will occur in the \(i\) th trial is \(p_{i}\). Let \(\mu\) be the number of occurrences of \(A\) in the first \(n\) independent trials. Find
(a) \(M \mu\),
(b) \(\mathbf{D}_{\mu}\),
(c) \(M\left(\mu-\sum_{1}^{n} p_{i}\right)^{3}\) and
(d) \(M\left(\mu-\sum_{1}^{n} p_{i}\right)^{4}\)