I have an example from a book that I dont quite understand, where they talk about drawing random samples from some domain, using a Power Distribution. With the domain being from 0-1.

Now, what I understand this to mean is that with a uniform probabilty distribution, the chances of picking any number from the domain are equal, so the pdf is just 1/domain. But with a Power Distribution pdf, then the probabilty of picking a number from the domain gets higher as the variable increases in value.

ie if i have the exponent (n), for this power distribution set to say 3,

the probability of picking 0.33 is 0.33^3

the probability of picking 0.78 is 0.78^3

But the pdf, needs to sum to 1 over the domain, so you need to add a normalization constant to do this, (c)

pdf = cx^n

With the restriction that the pdf has to sum over one, you integrate the above from 0-1 and work out the constant to be

c = n + 1

so now the pdf is

pdf(x) = (n + 1)x^n

But if i try this with examples, i get probabilities over 1.

n = 8

pdf(0.6) = (8+1)0.8^8 = 1.50

pdf(0.8) = (8+1)0.6^8 = 3.54

pdf(0.99) = (8+1)0.99^8 = 8.30