tailieunhanh - An Introduction to Financial Option Valuation Mathematics Stochastics and Computation_2

Tham khảo tài liệu 'an introduction to financial option valuation mathematics stochastics and computation_2', tài chính - ngân hàng, tài chính doanh nghiệp phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 3 Random variables OUTLINE discrete and continuous random variables expected value and variance uniform and normal distributions Central Limit Theorem Motivation The mathematical ideas that we develop in this book are going to involve random variables. In this chapter we give a very brief introduction to the main ideas that are needed. If this material is completely new to you then you may need to refer back to this chapter as you progress through the book. Random variables probability and mean If we roll a fair dice each of the six possible outcomes 1 2 . 6 is equally likely. So we say that each outcome has probability 1 6. We can generalize this idea to the case of a discrete random variable X that takes values from a finite set of numbers X1 X2 . xm . Associated with the random variable X are a set of probabilities P1 P2 . pm such that xi occurs with probability pi. We write P X xi to mean the probability that X xi . For this to make sense we require Pi 0 for all i negative probabilities not allowed 22m 1 pi 1 probabilities add up to 1 . The mean or expected value of a discrete random variable X denoted by E X is defined by m E X xipi. i 1 21 22 Random variables Note that for the dice example above we have E X 11 12 16 6 1 6 6 6 2 which is intuitively reasonable. Example A random variable X that takes the value 1 with probability p where 0 p 1 and takes the value 0 with probability 1 p is called a Bernoulli random variable with parameter p. Here m 2 xi 1 X2 0 P1 p and p2 1 p in the notation above. For such a random variable we have E X 1 p 0 1 p p. A continuous random variable may take any value in R. In this book continuous random variables are characterized by their density functions. If X is a continuous random variable then we assume that there is a real-valued density function f such that the probability of a X b is found by integrating f x from X a to X b that is P a X b ị f x dx. Here P a X b means the probability that a X b . For this