Binomial Distribution Review

A random variable \(K\) has a binomial distribution with parameters \(n\) and \(p\), denoted \(K \sim \text{Binom}(n, p)\), if its probability mass function is given by

\[ \begin{aligned} P(K = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, \ldots, n. \end{aligned} \]

The binomial distribution is often used to model the probability that a sequence \(n\) independent Bernoulli trials (yes/no experiments) where the probability of success is \(p\) will result in \(k\) total successes.

Part A

Please compute (by hand, showing your steps) the expectation \(\mathbb{E}[K]\).

Hint: Use linearity of expectation and indicator variables for each of the \(n\) independent Bernoulli trials.

Part B

The variance of a random variable \(K\) is given by \[ \begin{aligned} \text{var}(K) = \mathbb{E}[K^2] - \mathbb{E}[K]^2. \end{aligned} \]

In the case of the binomial distribution, the variance has formula \(\text{var}(L) = np(1-p)\).

Suppose that we let \(p = \frac{c}{n-1}\) for some constant \(c\). Please determine the limiting values of both the mean and variance of \(K\) in the limit as \(n\rightarrow \infty\).