C++ Neural Networks and Fuzzy Logic: Preface

Fuzzy Events, Means and Variances

Let us introduce you to fuzzy events, fuzzy means, and fuzzy variances. These concepts are basic to make a

study of fuzzy quantification theories. You will see how a variable’s probability distribution and its fuzzy set

are used together. We will use an example to show how fuzzy means and fuzzy variances are calculated.

Example: XYZ Company Takeover Price

Suppose you are a shareholder of company XYZ and that you read in the papers that its takeover is a prospect.

Currently the company shares are selling at $40 a share. You read about a hostile takeover by a group

prepared to pay $100 a share for XYZ. Another company whose business is related to XYZ’s business and has

been on friendly terms with XYZ is offering $85 a share. The employees of XYZ are concerned about their

job security and have organized themselves in preparing to buy the company collectively for $60 a share. The

buyout price of the company shares is a variable with these three possible values, viz., 100, 85, and 60. The

board of directors of XYZ have to make the ultimate decision regarding whom they would sell the company.

The probabilities are 0.3, 0.5, and 0.2 respectively, that the board decides to sell at $100 to the first group, to

sell at $85 to the second, and to let the employees buy out at $60.

Thus, you get the probability distribution of the takeover price to be as follows: