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Random variable with exponential distribution

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1. A single observation of a random variable having an exponential distribution with ----

If the null hypothesis is accepted if and only if the observed value of the random variable is less than 3.

a) Find the probabilities of Type I and Type II errors.

b) What is the power function of this test?

2. Let -- be a random sample from a distribution that is -- . Let the observed values be --. Assume the sample size --.
a) If -- is known, construct an exact 95% confidence interval for --
b) If -- is unknown but -- is known, construct an exact 95% confidence interval for --.

3. Let --- be a random sample with pdf ---, zero elsewhere, where -- is a positive real number.
a) Find the MLE, , of --
b) Find the MLE for the population median of the distribution.
c) Find the sufficient statistic for --.