1. Andy Guo 1 Handout Ch5(2) 實習

2. Andy Guo 2 Normal Distribution There are three reasons why normal distribution is important –Mathematical properties of the normal distribution have simple forms –Many random variables often have distributions that are approximately normal –Central limit theorem tells that many sample functions have distributions which are approximately normal The p.d.f. of a normal distribution

3. Andy Guo 3 Properties of Normal Distribution Proof: If we let y = (x–  )/ , then

4. Andy Guo 4 極座標 直角座標與極座標的轉換 ∫ ∫dxdy=rdrdθ X=r*cosθ ， y=r*sinθ Ex: x^2+y^2 ≦ 1 (x,y) r

5. Andy Guo 5 The m.g.f. of Normal Distribution

6. Andy Guo 6 Properties of Normal Distribution If the random variables X 1, …, X k are independent and if X i has a normal distribution with mean  i and variance  i 2, then the sum X 1 +... + X k has a normal distribution with mean  1 +... +  k and variance  1 2 +... +  k 2. Proof: The variable a 1 x 1 +... + a k x k + b has a normal distribution with mean a 1  1 +... + a k  k + b and variance a 1 2  1 2 +... + a k 2  k 2 Suppose that X 1, …, X n form a random sample from a normal distribution with mean  and variance  2, and let denote the sample mean. Then has a normal distribution with mean  and  2 /n.

7. Andy Guo 7 5.6.16 Suppose that the joint p.d.f. of two random variables X and Y is Find

8. Andy Guo 8 Solution

9. Andy Guo 9 Gamma Function For each positive number , let  (  ) be defined as: If , then  (  )= (  )  (  ) Proof: Let u= x  –1, v=  e – x then du= (  1) x  –2 and dv= e –x dx For every integer

10. Andy Guo 10 Gamma Distribution X has gamma distribution with parameters  and  (  and  >0) So

11. Andy Guo 11 Gamma Distribution For k = 1, 2, …, we have

12. Andy Guo 12 Gamma Distribution The m.g.f. of X can be obtained as: If X 1,..., X k are independent r.v. and if X i has a gamma distribution with parameters  i and , then the sum has a gamma distribution with parameters  and . Proof:

13. Andy Guo 13 Physical Meaning of Gamma Distribution When  is a positive integer, say  = n, the gamma distribution with parameters ( ,  ) often arises as the distribution of the amount of time one has to wait until a total of n events has occurred. Let X n denote the time at which the nth event occurs, then Notice that the number of events in [0, x] has a Poisson distribution with parameter  x, and  is the rate of events.

14. Andy Guo 14 Exponential Distribution A gamma distribution with parameters  = 1 and  is an exponential distribution. A random variable X has an exponential distribution with parameters  has: Memoryless property of exponential distribution

15. Andy Guo 15 Life Test Suppose X 1, …, X n denote the lifetime of bulb i and form a random sample from an exponential distribution with parameter . Then the distribution of Y 1 =min{X 1, …, X n } will be an exponential distribution with parameter n . Proof: Determine the interval of time Y 2 between the failure of the first bulb and the failure of a second bulb. –Y 2 will be equal to the smallest of (n  1) i.i.d. r.v., so Y 2 has an exponential distribution with parameter (n  1) . –Y 3 will have an exponential distribution with parameter (n  2) . –The final bulb has an exponential distribution with parameter .

16. Andy Guo 16 Physical Meaning of Exponential Distribution Following the physical meaning of gamma distribution, an exponential distribution is the time required to have for the 1st event to occur, i.e., where  is rate of event. In a Poisson process, both the waiting time until an event occurs and the period of time between any two successive events will have exponential distributions. In a Poisson process, the waiting time until the nth occurrence with rate  has a gamma distribution with parameters n and .

17. Andy Guo 17 Example 5.9.1: Radioactive Particles For the Poisson process example in Example 5.4.3, suppose that we are interested in how long we have to wait until a radioactive particle strikes our target. Let Y 1 be the time until the first particle strikes the target, and X be the number of particles that strike the target during a time period of length t. X has a Poisson distribution with mean t, where is the rate of the process. The c.d.f. of Y 1 = F(y) = (for t > 0) Taking the derivative of the c.d.f. we obtain: The p.d.f. of Y 1 = f(y) = (for t > 0)

18. Andy Guo 18 5.9.22 Consider the Poisson process of radioactive particle hits in Example 5.9.1. Suppose that the rate of the Poisson process is unknown and has a gamma distribution with parameters and. Let X be the number of particles that strike the target during t time units. Prove that the conditional distribution of given X=x is a gamma distribution, and find the parameters of that gamma distribution

19. Andy Guo 19 Solution

20. Andy Guo 20 補充 Let X be a random variable for which the p.d.f. is f and for which Pr(a<X<b)=1. Let Y=r(x), and suppose that r(x) is continuous and either strictly increasing or strictly decreasing for a<x<b. Suppose also that a<X<b if only if α<Y<β. Then the p.d.f. g of Y is specified by the relation

21. Andy Guo 21 Example f(x)=2x,for0<x<1, 求 g(y) Solution: y=2x =>x=y/2, s(y)=y/2 ds(y)/dy=1/2 所以 g(y)=f(y/2)*(1/2)=y/2 for 0<y<2