Assignment - Statistical Theory IV

Goal: This course is designed to provide the student with an enhanced knowledge of theories of statistics

General Objectives: On completion of this course, the diplomate should be able to:

1. Understand distributions of independent random variables.

2. Understand various distributions related to the normal

3. Understand Cochran’s theorem

4. Understand the Neyman/Pearson lemma for testing of hypothesis

5. Understand the methods of maximum likelihood estimation

6. Understand the method of minimum variance unbiased estimation

1.1 Define a necessary and sufficient condition for the independent of two discrete variables.

1.2 Derive the characteristic function of the sum of independent variables

1.3 Derive the distribution function of the sum of two independent random variables

2.1 Define the gamma function

2.2 Define and derive the probability density function of the Χ² distribution

2.3 Derive the characteristic function of the Χ² distribution

2.4 Explain the concept of degrees of freedom

2.5 Compute the first and the second moments of the Χ² distribution

2.6 Define and derive the students t distribution

2.7 Compute the first and the second moments of the t distribution

2.8 Define and derive the Fisher’s F distribution

2.9 Compute the first and the second moments of the r distribution.

3.1 State Cochran’s theorem for K samples

3.2 Apply Cochran’s theorem to samples from normal populations

8.1 Define the test of a simple hypothesis against a simple alternative hypothesis.

8.2 Distinguish between randomized and nonrandomized tests

8.3 Define the power of a test and the UMP tests

8.4 Derive and represent OCcurves

8.5 State and prove Neyman /Pearson lemma to find the most powerful test

5.1 Define and compute the likelihood function of random variables

5.2 Define and compute the maximum likelihood estimators of parameters of the normal, poisson and the binomial distributions.

6.1 Define and compute Crammer-Rao bounds

6.2 Define and compute Bhattacharya bounds for estimators of parameters of the normal, poisson and binomial distribution

6.3 Define and compute unbiased estimators and the MVUE parameters of the distribution in 6.2

6.4 Define and illustrate sufficient statistics and complete statistics

6.5 State and prove the Rao-Blackwell theorem

6.6 Apply the Rao-Blackwell theorem to solve problems.