Solved – Since the beta distribution is similar in form to the binomial, why do we need the beta distribution

They're related, but not actually so similar in form.

In the beta, the variable (and its complement) is raised to some power, but in the binomial the variable is the power (and it also appears in a binomial coefficient).

While the functional forms do look somewhat alike (there are terms in one that correspond to terms in the other), the variables that represent the parameters and the random variable in each are different. That's rather important; it's why they're actually not the same thing at all.

The binomial distribution is usually used for counts, or in scaled form, for count-based proportions (though you could use it for other bounded discrete random variables on a purely pragmatic basis). It's discrete.

The beta distribution is continuous, and so is not normally used for counts.

By way of example, compare these two functions:

$y = b^x,, x=0,1,2,3,…$ and $y = x^a,, 0<x<1$.

Both these functions are defined by expressions of the same form (something of the form $c^d$), but the roles of variable and constant are interchanged and the domain is different. The relationship between the beta and the binomial is like the relationship between those two functions.

– In summary: different form, and different domain

Here's a simple example of a beta distribution, the $text{beta}(1,1)$. Which binomial distribution does the same job?

Or consider a $text{beta}(2,1)$; it's hard to find a binomial which looks similar. Here's one attempt:

The entire beta pdf sits between the first two green spikes in the binomial pf, though they can't really be shown on the same plot because the y-axes measure different things.

While the shapes are vaguely similar in the sense that they're both left skew, they're really quite different, and used for different things.

—

Here's a challenge:

For $X_1simtext{beta}(1,1)$ and a $X_2simtext{beta(3,2)}$, find binomial distributions (presumably scaled) that can simultaneously reasonably accurately (say to within $c=(0.95, 1.05)$ times the correct probability, give or take) which have the same mean and variance or mean and range (you pick), but also approximately reproduce the probability of being in these three subintervals: (a) $(1/pi,1/e)$, (b) $(exp(-frac{1}{2}),2/pi)$, and (c) $(exp(-3),1/pi^2)$

The beta is used to do many things, including model continuous proportions, act as a prior on the $p$ parameter of a binomial, it is the distribution of uniform order statistics (and can be used in the derivation of the distribution of order statistics for other continuous distributions, used as a mixing distribution for the binomial $p$ (producing the beta-binomial distribution), to model task completion times in project management, and many other things.