MTH136/STA114: Statistics

Homework #3

Due: Wednesday, Feb 7, 2001

1. The random variables X₁, ..., X_n are independent and uniformly distributed on the interval from 0 to theta, with theta unknown. Find a one-dimensional Sufficient Statistic for theta, i.e., a real-valued function T(X₁, ..., X_n) such that the likelihood function for theta depends on the data only through T. Give the likelihood function, too.
2. The random variables X₁, ..., X_n are independent with the Exponential distribution with rate lam, i.e., with density function

   f(x_j|lam) = lam e^{-lam x_j}

   1. Find the Likelihood function L(lam) for n independent observations
   2. Find the Maximum Likelihood Estimator lam-hat that maximizes the L(lam)
   3. What is the exact probability distribution of the numerical inverse 1/lam-hat?
   4. By the Central Limit Theorem, this distribution is approximately normal if n is large enough. What are its (approximate or exact) mean and variance?
3. DeGroot Chapter 6 Section 3-6 problems are postponed to next week. Cheers, -R