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Relation between Beta(a,b) and U(0,1)
Beta distribution and standard uniform distribution which is a special case of uniform distribution with 0,1 has some relationships.
Firstly, Beta(1,1) and U(0,1) look exactly the same distributions and secondly, order statistics of U(0,1) looks same with beta distribution with specific parameters.
Firstly, let's take a look at basic information of two distributions.
<Beta distribution>
You should memorize that range of random variable of Beta distribution should be between 0 and 1 because it notes the probability.
<Uniform distribution>
<Standard Uniform distribution>
Relationship 1
Beta distribution with a=1 and b=1 becomes the same distribution with Standard Uniform distribution.
Relationship 2
Order statistics of Standard Uniform distribution has relations with Beta distribution.
Specifically, 'k'th order statistics of Standard Uniform distribution is same with Beta(k,n-k+1)
The first image is about the way to find pdf of order statistics which is needed to find pdf of order statistics of standard uniform distribution.
You can find the way to find pdf of 'k'th order statistics of standard uniform dritribution.
'k'th order statistics of U(0,1) is same with Beta(k,n-k+1) distribution even the range of random variable.
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