tailieunhanh - Probability and Statistics

Finance has become increasingly more quantitative, drawing on techniques in probability and statistics that many finance practitioners have not had exposure to before. In order to keep up, you need a firm understanding of this discipline. Probability and Statistics for Finance addresses this issue by showing you how to apply quantitative methods to portfolios, and in all matter of your practices, in a clear, concise manner. Informative and accessible, this guide starts off with the basics and builds to an intermediate level of mastery. . | Chapter 1 Discrete Probability Distributions Simulation of Discrete Probabilities Probability In this chapter we shall first consider chance experiments with a finite number of possible outcomes U 1 Cc 2 cưn. For example we roll a die and the possible outcomes are 1 2 3 4 5 6 corresponding to the side that turns up. We toss a coin with possible outcomes H heads and T tails . It is frequently useful to be able to refer to an outcome of an experiment. For example we might want to write the mathematical expression which gives the sum of four rolls of a die. To do this we could let Xi i 1 2 3 4 represent the values of the outcomes of the four rolls and then we could write the expression x4 x2 x3 x4 for the sum of the four rolls. The Xj s are called random variables. A random variable is simply an expression whose value is the outcome of a particular experiment. Just as in the case of other types of variables in mathematics random variables can take on different values. Let X be the random variable which represents the roll of one die. We shall assign probabilities to the possible outcomes of this experiment. We do this by assigning to each outcome ajj a nonnegative number in such a way that m wi m w2 4----1- 1 . The function is called the distribution function of the random variable X. For the case of the roll of the die we would assign equal probabilities or probabilities 1 6 to each of the outcomes. With this assignment of probabilities one could write 2 P X 4 1 2 CHAPTER 1. DISCRETE PROBABILITY DISTRIBUTIONS to mean that the probability is 2 3 that a roll of a die will have a value which does not exceed 4. Let Y be the random variable which represents the toss of a coin. In this case there are two possible outcomes which we can label as H and T. Unless we have reason to suspect that the coin comes up one way more often than the other way it is natural to assign the probability of 1 2 to each of the two outcomes. In both of the above experiments each outcome is .