Abstract:
Here we are interested in exploring further the Wilson confidence intervals(CI). In particular the aim is to develop Wilson intervals for random variables that have variances which are a quadratic function of the mean of the random variable, θ = E[X]. Example of this type of random variable include the Poisson, Binomial, Beta-binomial. Having derived a Wilson CI for the generic case, say V[X] = a + bθ +cθ2, then we will compare how this new interval performs relative to standard CI approaches using Monte Carlo simulation. Particular interest will be paid to more commonly used distributions such as the Poisson and Gamma.

Contents:
1. Generalized Wilson Confidence Interval
1.1 Introduction
1.2 Monte Carlo Approach
1.3 Summary of Processes

2. Finding a Generalized Form for the Wilson Confidence Inerval
2.1 Wilson Confidence Inerval

3. The Uniform Distribution
3.1 Finding Quadratic Variance for the Uniform Distribution
3.2 Comparing Wilson Method to Traditional
3.3 Summary of Results

4. The Poijsson Distribution
4.1 Finding a Quadratic Variance for Poisson
4.2 Comparison of Wilson Against Other Methods
4.3 Summary of Results

5. The Expoential Distribution
5.1 Finding Quadratic Variance for the Exponential Distribution
5.2 Comparing Wilson Method to Traditional
5.3 Summary of Results

6. The Gamma Distribution I
6.1 Finding Quadratic Variance
6.2 Comparing Wilson Mthod to Traditional
6.3 Summary of Resuts

7. The Gamma Distribution II
7.1 Finding Quadratic Variance for the Gamma Distribution
7.2 Comparing Wilson Method to Traditional
7.3 Summary of Results

8. Conclusions