The standard way to define the Beta is using the following pdf:

Where $ x \in [0,1]$ and $B( \alpha, \beta ) $ is the Beta Function:

When we use this parameterisation, the first two moments are:

Mean:

$$\hat{\beta} = (1- m) \Bigg( \frac{m (1-m) }{v} - 1 \Bigg) $$

Define.....

And then Mean = blah

Variance = blah...

**Construction of Probability Distributions**

I always enjoyed thinking about how some of the common probability distributions

We can think of the construction as being based on an interesting integration, along with a normalisation constant.

$$c \int_{0}^\infty (1-x)^{(\beta - 1)} x^{(\alpha -1)} dx$$

Then the requirement that the integral integrate to 1 is equivalent to:

$$c = \frac{1}{\int_{0}^\infty (1-x)^{(\beta - 1)} x^{(\alpha -1)}} dx$$

i.e. $c = \frac{1}{B ( \alpha, \beta )}$

Isn't this cool? In order to ensure that the overall probability sums to 1, we are effectively partitioning the.

In fact that is quite a common construction for probability distributions.

Euler integrals:

Euler integral of the first kind:

Beta dist

Euler integral of the second kind:

$$ \Gamma ( z ) = \int_{0}^\infty x^{(z - 1)} e^{-x} dx$$

The Poisson distribution can also be seen as an infinite sum and a normalisation constant:

Take the definition of the exponential function:

We can use the same construction here:

$ f(x) =