Credit card fraud is perpetrated through stolen credit cards or credit card info

Question

Credit card fraud is perpetrated through stolen credit cards or credit card information. For years, credit card issuers have been using data mining and statistical tools to detect fraud. Citibank reported that knowing the type of product or service bought, frequency of purchases, and size and location of transaction can significantly reduce fraud. (Source: Jesus Mena, Investigative Data Mining for Security and Criminal Detection, Butterworth-Heinemann, pp. 250-251) A data-mining analyst at a major credit card company would like to construct and test a simple logistic regression model for detecting credit card fraud using data on card transactions classified as either fraudulent or non-fraudulent. The dependent variable for the model is: y odds ratio that the transaction is due to credit card fraud = The independent variables for the model are chosen from the following: X1 = dollar amount of the transaction X2 = number of transactions in the preceding 12 hours I3-1 if the Standard Industry Code (SIC) for the product or service bought has never appeared in the card owner's transaction history; 0 otherwise 1 if the postal code of the transaction has never appeared in the card owner's transaction history; 0 otherwise The analyst would like to test a logistic regression model that predicts credit card fraud using the dollar amount of the transaction, the number of transactions in the preceding 12 hours, and the indicator variable for whether the postal code of the transaction never appeared in the card owner's transaction history. (Note that actual fraud-detection models used by credit card companies are much more complicated than the above, including up to hundreds of independent variables.) Let F denote the event that the transaction is due to credit card fraud. The estimated probability of event F is: O exp(bo bixi b2x2 b4l4) O exp(bobixib2x2bal4)/[1exp(bo+bixi b2x2bal4) O exp(Bo + Pal3 + 414) / [1 + exp(Bo + 313 + Bald

Explanation / Answer

Question 1

Here the probability F = exp (b0 + b1x1 + b2x2 + b4I4)/ [1 + exp (b0 + b1x1 + b2x2 + b4I4)]

Y = e(b0 + b1x1 + b2x2 + b4I4)

so for x1 = $ 412 ad x2 = 4 ; SIC =1

Y1 = e(0.0297 + 0.0045 * 412 + 0.2701 * 4 + 0.1645) = 22.8419

and Probability F1 = e3.1286 / (1 + e3.1286) = 0.958

when x2 = $ 413

Y2 = e(0.0297 + 0.0045 * 413 + 0.2701 * 4 + 0.1645) = 22.9450

Probability F1 = e3.1331 / (1 + e3.1331) = 0.95824

Ratio of the second odds to the first is = 22.9450/22.8419 = 1.0045, whish is equal to exp(b1) = exp(0.0045) = 1.0045

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