Mean and standard deviation for a binary random variable One of the simplest discrete random variables is a binary random variable (sometimes called a Bernoulli random variable). It takes on only two distinct values, 1 and 0 , with probabilities \(p\) and \(1-p\), respectively. Let \(X\) denote a binary random variable, with possible values \(x=1\) or 0 and \(\mathrm{P}(1)=p\) and \(\mathrm{P}(0)=1-p\).

a. Show that the mean of \(X\) is equal to \(p\).

b. Show that the variance of \(X\) is equal to \(p(1-p)\) and the standard deviation is equal to \(\sqrt{p(1-p)}\).

c. Show that the mean and standard deviation result from setting \(n=1\) in the formulas for the mean and standard deviation of the binomial distribution.