Probability and Statistics
APPENDIX A Probability and Statistics The basics of probability and statistics are presented in this appendix, provid- ing the reader with a reference source for the main body of this book without interrupting the flow of argumentation there with unnecessary excursions. Only the basic definitions and results have been included in this appendix. More complicated concepts (such as stochastic processes, Ito Calculus, Girsanov Theorem, etc.) are discussed directly in the main body of the book as the need arises. A.1 PROBABILITY, EXPECTATION, AND VARIANCE A variable whose value is dependent on random events, is referred to as a random variable or a random number. An example of a random variable is the number observed when throwing a dice. The distribution of this ran- dom number is an example of a discrete distribution. A random number is discretely distributed if it can take on only discrete values such as whole numbers, for instance. A random variable has a continuous distribution if it can assume arbitrary values in some interval (which can by all means be the infinite interval from −∞ to ∞). Intuitively, the values which can be assumed by the random variable lie together densely, i.e., arbitrarily close to one another. An example of a discrete distribution on the interval from 0 to ∞ (infinity) might be a random variable which could take on the values 0.00, 0.01, 0.02, 0.03, ..., 99.98, 99.99, 100.00, 100.01, 100.02, ..., etc., like a Bund future with a tick-size of 0.01% corresponding to a value change of 10 euros per tick on a nominal of 100,000 euros.
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