Statistical Theory III

Goal: This course is designed to introduce students to distributing continuous types and to elementary estimation theory

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

1. Understand distributions of the continuous type.

2. Understand the concept of the use of conditional distributions.

3. Understand the distribution of functions of random variables

4. Understand further uses of the central limit theorem.

5. Understand the bivariate normal distribution

6. Understand the concept of the Chebyshev inequality and its uses

7. Understand the method of least squares estimation

1.1 Define continuous random variables.

1.2 Define the probability distribution function of a continuous variable.

1.3 Evaluate the probability distribution function of a continuous variable.

1.4 Define the distribution function of a continuous random variable.

1.5 Determine the distribution function of a continuous random variable using the probability distribution function.

1.6 Evaluate the expected value of a continuous random variable.

1.7 Evaluate the moment generating function of a variable.

1.8 Evaluate the characteristic function of a variable

2.1 Define conditional probability density function of  X given Y.

2.2 Compute conditional probability such as  P(X/Y)=y.

2.3 Define conditional mean of X and the conditional variance of X given Y.

3.1 Define the distribution of functions of random variables.

3.2 Determine the mean, the variance and moment generating fFunction of a function such as  Y=(X1, X2).

3.3 Identify functions that are linear combinations of random variables.

3.4 Calculate the expected values and variances of the function in 3.3 above.

3.5 Find the moment generating functions and the distributions of the sum of independent random variables.

4.1 Review the central limit theorem.

4.2 State the importance of the central limit theorem.

4.3 Approximate probabilities when n is  “sufficiently large” using the central limit theorem.

5.1 Define the bivariate normal distribution.

5.2 Derive the moment generating function of the bivariate normal distribution.

5.3 Obtain the marginal and the conditional densities of the bivariate normal distribution

6.1 State the Chebyshev Inequality.

6.2 Prove the law of large numbers applying the Chebyshev Inequality.

6.3 Solve some problems using the inequality

7.1 Distinguish between point and estimate intervals.

7.2 Define the least squares estimator.

7.3 Define the best linear unbiased estimator (BLUE).

7.4 State the Gauss-Markov theorem.

7.5 Obtain the least squares estimates of βo and β1 in the model y=β0 + β1X + E

7.6 State and explain the desirable properties of a good estimator unbiasedness, efficiency, sufficiency and consistency.