Let {N(t):t > 0} be a Poisson process with intensity 1 > 0 (1) Find P(N(t) = n).
(2) For x > 0 and t > 0, calculate P(N(t + x) – N(t) = 0).
(3) Find P(Y(t) > x), where Y (t) is the excess at t.
(4) Find P(A(t) > x) for 0 < x < t, where A(t) is the age at t.
Competitive Auctions on eBay. com. The file eBayAuctions .xls contains in formation on 1972 auctions that transacted on eBay. com during May-June 2004. The goal is to use these data to build a model that will classify auctions as competitive or non- competitive. A competitive auction is defined as an auction with at least two bids placed on the item auctioned. The data include variables that describe the item (auction category). the seller (his/her eBay rating), and the auction terms that the seller selected (auction duration, opening price, opening price, currency, day-of-week of auction close). In addition, we have the price at which the auction closed. The task is to predict whether or not the auction will be competitive.
Data Preprocessing. Create dummy variables for the categorical predictors. These include Category (18 categories), Currency (USD, GBP, euro), EndDay (Monday-Sunday). and Duration (1, 3,5,7, or 10 days). Split the data into training and validation datasets using a 60 : 40% ratio. A. Fit a classification tree using all predictors, using the best-pruned tree. To avoid overfitting, set the minimum number of records in a terminal node to 50. Also set the maximum number of levels to be displayed at seven (the maximum allowed in XLMiner). Write down the results in terms of rules. (Note: If you had to slightly reduce the number of predictors due to software limitations, or for clarity of presen- tation, which would be a good variable to choose?)
B. Is this model practical for predicting the outcome of a new auction?
C. Describe the interesting and uninteresting information that these rules provide
D. Fit another classification tree (using the best-pruned tree, with a minimum number of records per terminal node = 50 and maximum allowed number of displayed level), this time only with predictors that can be used for predicting the outcome of a new auction. Describe the resulting tree in terms of rules. Make sure to report the smallest set of rules required for classification.
E. Plot the resulting tree on a scatter plot: Use the two axes for the two best (quantitative) predictors. Each auction will appear as a point, with coordinates corresponding to its values on those two predictors. Use different colors or symbols to separate competitive and noncompetitive auctions. Draw lines (you can sketch these by hand or use Excel) at the values that create splits. Does this splitting seen reasonable with respect to the meaning of the two predictors? Does it seem to do a good job of separating the two classes?
F. Examine the lift chart and the classification table for the tree. What can you say about the predictive performance of this model?
G. Based on this last tree, what can you conclude from these data about the chances of an auction obtaining at least two bids and its relationship to the auction settings set by the seller (duration, opening price, ending day, currency)? What would you recommend for a seller as the strategy that will most likely lead to a competitive auction?