2. 2.4 Continuous r.v.

3. Suppose that F(x) is the distribution function of r.v. X ， if there exists a nonnegative function f(x) ， (-  <x<+  ) ， such that for any x ， we have Definition2.8---P35 The function f(x) is called the Probability density function （ pdf ） of X, i.e. X ～ f(x), (-  <x<+  ) The geometric interpretation of density function

4. 1 (1) and (2) are the sufficient and necessary properties of a density function Note: Properties of f(x)-----P35 Suppose that the density function of X is specified by Try to determine the value of K.

5. （ 3 ） For any a ， if X ～ f(x) ， (-  <x<  ) ， then P{X=a} ＝ 0 。 Proof P36

6. 1 Proof P35

7. (5) If x is the continuous points of f( x ), then P35 Note:P36---(1)

8. Example1 Suppose that the density function of X is specified by Try to determine 1)the value of K 2)the d.f. F(x), 3)P(1<X≤3.5) 4)P(x=3) 5)P(x>3.5 ∣ x>3)

9. Suppose that the distribution function of X is specified by Try to determine (1) P{X<2},P{0<X<3},P{2<X<e-0.1}. (2)Density function f(x) Example2

10. 1.Uniformly distribution if X ～ f(x) ＝ It is said that X are uniformly distributed in interval (a, b) and denote it by X~U(a, b) For any c, d (a<c<d<b) ， we have Several Important continuous r.v.

11. Example 2.14-P38

12. 2. Exponential distribution If X ～ It is said that X follows an exponential distribution with parameter >0, the d.f. of exponential distribution is

13. Example Suppose the age of a electronic instrument 电子仪器 is X (year), which follows an exponential distribution with parameter 0.5, try to determine (1)The probability that the age of the instrument is more than 2 years. (2)If the instrument has already been used for 1 year and a half, then try to determine the probability that it can be use 2 more years.

15. The normal distribution are one the most important distribution in probability theory, which is widely applied In management, statistics, finance and some other areas. 3. Normal distribution A B Suppose that the distance between A ， B is  ， the observed value of  is X, then what is the density function of X ？

16. where  is a constant and  >0 ， then, X is said to follows a normal distribution with parameters  and  2 and represent it by X ～ N( ,  2 ). Suppose that the density function of X is specified by

17. (1) symmetry the curve of density function is symmetry with respect to x=  and f(  ) ＝ max f(x) ＝. Two important characteristics of Normal distribution

18. (2)  influences the distribution  ， the curve tends to be flat ，  ， the curve tends to be sharp ，

19. 4.Standard normal distribution A normal distribution with parameters  ＝ 0 and  2 ＝ 1 is said to follow standard normal distribution and represented by X~N(0, 1) 。

20. and the d.f. is given by the density function of normal distribution is

21. The value of  (x) usually is not so easy to compute directly, so how to use the normal distribution table is important. The following two rules are essential for attaining this purpose. Note ： (1)  (x) ＝ 1 －  ( － x) ； (2) If X ～ N( ,  2 ) ， then

22. 1 X~N(-1,2 2 ), P{-2.45<X<2.45}=? 2. X  N( ,  2 ), P{  -3  <X<  +3  }? EX 2 tells us the important 3  rules, which are widely applied in real world. Sometimes we take P{|X-  |≤3  } ≈1 and ignore the probability of {|X-  |>3  }

23. Example The blood pressure of women at age 18 are normally distributed with N(110,12 2 ).Now, choose a women from the population, then try to determine (1) P{X x}<0.05

25. Example 2.15,2.16,2.17,2.18-P40-42

26. Homework: P50--- Q15 ， 18 P51: 17,19,