Chapter 5 Continuous Random Variables
Chapter 5 Continuous Random Variables
5.1 Introduction Examples: (1)The time that a train arrives at a specified stop. (2)The lifetime of a transistor
5.1 Introduction Examples: (1) The time that a train arrives at a specified stop. (2) The lifetime of a transistor
汝Definition We say that X is a continous random variables if there exsists a nonnegative funtions f,defined for all real xE(-,+o),having the property that for any set B of real numbers P({X∈B)=∫f(x) The funtionf is called the probability density funtion of the random var iable X
Definition , ( , ), ({ We say that X is a random variables if there exsists a nonnegative funtions f defined for all real x having the property that for any set B of real num co ber ntinous s P ∈ −∞ +∞ }) ( ) var . B probab X B f x dx The funtion f is called the ility density funt of t io he random i X n able ∈ = ∫
*Important results (1)1=PiXe(0,)=f(x)dx (2)P{a≤X≤b}=∫f(x)d (3)P{X=a}=f(x)d=0 (4)P(X<a=P{Xsa)=F(a)="f(x)d
Important results (1)1 { ( , )} ( ) P X f x dx ∞ −∞ = ∈ −∞ ∞ = ∫ (2) { } ( ) b a P a X b f x dx ≤≤= ∫ (3) { } ( ) 0 a a P X a f x dx = = = ∫ (4) { } { } ( ) ( ) a PX a PX a Fa f xdx −∞ <= ≤= = ∫
Example 1a Suppose that X is a continous random variable whose probability density function is given by f(x)= C(4x-2x2)0<x<2 0 otherwise (a)what is the value of C? (b)Find P(X>1)
Example 1a 2 (4 2 ) 0 2 () 0 () ? ( ) ( 1) Suppose that X is a continous random variable whose probability density function is given by Cx x x f x otherwise a what is the value of C b Find P X − << = >