1. ## Confidence Interval

Construct a symmetric two-sided $(1-\alpha)100\%$ confidence interval for the unknown parameter $\beta > 0$ in a $Beta(1,\beta)$ sample.

The hint is to find the MLE of $\beta$ and determine the distribution of $-log(1-X)$ for a $Beta(1,\beta)$ random variable $X$

2. Write your likelihood function, take the logarithm and then differentiate to find the MLE of beta.

3. I'm trying to write out the loglikelihood function for $Beta(1,\beta)$

I simply don't understand the pdf of the beta distribution. Its pdf is $\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}$ In my case, it would be $\frac{(1-x)^{\beta-1}}{B(1,\beta)}$

It's using the definition to define itself, how does this make any sense?

4. use $x_i$ and take the product from 1 to n
and write the constants with respect to the gamma function.