Homework+4

Homework 4 (Problems) (GWU EMSE-271)
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 * Notes for Homework Solutions Discussion In Class**:

1a.
 * "Created" distribution
 * Note range is 0 to 1.
 * He checked to be sure f(x|theta) is proper density.
 * Compute E[X[ by integrating, note have to multiple first by x. Manipulates to get density function, trick to let it integrate to 1.
 * Solved treating Xbar as a random variable, (CAP)
 * Then substitute xbar to get MOM.

1b.
 * For MLE always formulate as log of product.
 * Instead of __, maximixe the derivative of the log (ln in this case)
 * Set equal to zero and solve
 * Used excel functions to compute (Homework 4-1 Solutions)
 * Shows what the function looks like using the data table approach in excel.
 * Select calculations want to perform, columns: x, likelihood and log-likelihood
 * Nice to plot to see there is only one maximum
 * Good if concave. If not, the there are only local maximums, minimums or saddles.
 * Optimize using the log-likelihood because it is better behaved

2a.
 * Derived the exponential inverse equations, since it would be needed later
 * Used (Phil) E[X] and formulas on Slide 47 to compute lamda

2b. 90% credibility interval
 * Used inverse equation to compute 5th and 95th percentile values

2c. MOM and MLE Chi-squared tests
 * Used bin beginning at 0 and infinity, not minimum and maximum or you don't get 50?? data values. Function is CDF from 0 not 1.6.
 * MOM: Talked about requirements. Ei in each bin are >3 (3, 4, 5 are what some suggest)
 * Can combine two bins to make the numbers guidelines work.
 * But not meeting guidelines of 5-10 bins for 50 data points.
 * MLE:

2d. Plotted credibility intervals by inserting in the 5th and 95th percentile lines.
 * Big point
 * Minitab plots using AD. AD = xxx is the AD statistic, like in our Excel Chi-Squared exercise, the Chi-Squared statistic. The P-Value on that page is the P-Value associated with the AD statistic.

?? P-Value and Credibility or Confidence Intervals


 * Sources:**
 * EMSE 271, Fall 2009