1. Confidence Interval

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

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

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 $\displaystyle Beta(1,\beta)$

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

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

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

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