[理學(xué)]概率論與數(shù)理統(tǒng)計(jì)英文第三章-展示頁
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【正文】 of is (Figure ) , . Figure Probability distribution in Example Question. What is the difference between distribution functions and probability distributions例2 有一種驗(yàn)血新方法：把k個(gè)人的血混在一起進(jìn)行化驗(yàn)，如果結(jié)果是陰性，那么對(duì)這k 個(gè)人只作一次檢驗(yàn)就夠了，如果結(jié)果是陽性，那么必須對(duì)這k個(gè)人再逐個(gè)分別化驗(yàn)，這時(shí)k個(gè)人共需作k+1次檢驗(yàn)。twopoint distribution(兩點(diǎn)分布) X01P1pp某學(xué)生參加考試得5分的概率是p, X表示他首次得5分的考試次數(shù)，求X的分布。 3. Random Variables Definition of Random VariablesIn engineering or scientific problems, we are not only interested in the probability of events, but also interested in some variables depending on sample points. （定義在樣本點(diǎn)上的變量）For example, we maybe interested in the life of bulbs produced by a certain pany, or the weight of cows in a certain farm, etc. These ideas lead to the definition of random variables.1. random variable definitionDefinition A random variable is a real valued function defined on a sample space。 . it assigns a real number to each sample point in the sample space.Here are some examples.Example A fair die is tossed. The number shown is a random variable, it takes values in the set . Example The life of a bulb selected at random from bulbs produced by pany A is a random variable, it takes values in the interval . Since the outes of a random experiment can not be predicted in advance, the exact value of a random variable can not be predicted before the experiment, we can only discuss the probability that it takes some value or the values in some subset of R.2. Distribution functionDefinition Let be a random variable on the sample space . Then the function . is called the distribution function of Note The distribution function is defined on real numbers, not on sample space.Example Let be the number we get from tossing a fair die. Then the distribution function of is (Figure ) Figure The distribution function in Example 3. PropertiesThe distribution function of a random variable has the following properties:(1) is nondecreasing.In fact, if , then the event is a subset of the event ,thus (2), .(3)For any , .This is to say, the distribution function of a random variable is right continuous.Example Let be the life of automotive parts produced by pany A , assume the distribution function of is (in hours)Find ,.Solution By definition, . Question： What are the probabilities and ?Example A player tosses two fair dice, if the total number shown is 6 or more, the player wins $1, otherwise loses $1. Let be the amount won, find the distribution function of .Solution Let be the total number shown, then the events contains sample points, . Thus , And so Thus Figure The distribution function in Example The distribution function of random variables is a connection betweenprobability and calculus. By means of distribution function, the main tools in calculus, such as series, integrals are used to solve probability and statistics problems. Discrete Random Variables 離散型隨機(jī)變量In this book, we study two kinds of random variables.Definition A random variable is called a discrete random variable, if it takes values from a finite set or, a set whose elements can be written as
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