In a binomial setting, the random variable, X, that gives the number of successes is binomially distributed. The sample proportion can then be calculated as $\frac{X}{n}$ where $n$ is your sample size. My textbook states that
This proportion does not have a binomial distribution
however since $\frac{X}{n}$ is simply a scaled version of a binomially distributed random variable $X$, shouldn't it also have a binomial distribution?
Best Answer
As you state, the sample proportion is a scaled binomial (under a few assumptions). But a scaled binomial is not a binomial distribution; a binomial can only take on integer values, for example. Of course, it is very easy to figure out the pmf, cdf, expected value, variance, etc. from what we know of the binomial distribution, which I think is what you're getting at. But if you were to say something like "the sample proportion is a binomial, so the expected value is $np$, as is for all binomials", you would be clearly wrong.
If you wanted to be really technical, if $n$ = 1, then the sample proportion is still a binomial distribution.