概率論与数理统计(英文)第四章.doc
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4. Continuous Random Variable 连续型随机变量
Continuous random variables appear when we deal will quantities that are measured on a continuous scale. For instance, when we measure the speed of a car, the amount of alcohol in a persons blood, the tensile strength of new alloy.
We shall learn how to determine and work with probabilities relating to continuous random variables in this chapter. We shall introduce to the concept of the probability density function.
4.1 Continuous Random Variable
1. Definition
Definition 4.1.1 A function f(x) defined on is called a probability density function (概率密度函数)if:
(i) ;
(ii) f(x) is intergrable (可积的) on and .
Definition 4.1.2
Let f(x) be a probability density function. If X is a random variable having distribution function
, (4.1.1)
then X is called a continuous random variable having density function f(x). In this case,
. (4.1.2)
2. 几何意义
3. Note
In most applications, f(x) is either continuous or piecewise continuous having at most finitely many discontinuities.
Note 1 For a random variable X, we have a distribution function. If X is discrete, it has a probability distribution. If X is continuous, it has a probability density function.
Note 2 Let X be a continuous random variable, then for any real number x,
.
4. Example
Example 4.1.2
Find k so that the following can serve as the probability density of a continuous random variable:
Solution To satisfy the conditions (4.1.1), k must be nonnegative, and to satisfy the condition (4.1.2) we must have
so that .
(Cauchy distribution 柯西分布)
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Example 4.1.3 Calculating probabilities from the probability density function
If a random variable has the probability density
Find the probability from that it will take on value
(a) between 0 and 2;
(b) greater than 1.
Solution Evaluating the necessary integrals, we get
(a)
(b)
Example 4.1.4
Determining the distribution function of X,
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