苏良军高等数理统计习题答案.pdfVIP

Advanced Mathematical Statistics Chapter 1 1 Prove the following: c c (a) If the events A and B are independent, then so are A and B . (b) The events Ωand ∅are independent of any event A. (c) Is there any event which is independent of itself? If so, give an example. Answer. c c c (a) The stated result is immediate from P (A ∩B ) = P ((A∪B) ) = 1 −P (A∪B) = 1 −P (A) − P (B) + P (A∩B) = 1 −P (A) −P (B) + P (A) P (B) = (1 −P (A)) (1 −P (B)) = P (Ac c ) P (B ) . (b) For any event A, we have P (A∩Ω) = P (A) = P (A) P (Ω) , and P (A∩∅) = P (∅) = P (A) P (∅) = 0,as desired. (c) Yes. From the definition of independence, an event A is independent of itself if P (A∩A) = P (A) P (A) ,i.e., P (A) = 1,or 0. Thus, Ωand ∅are obvious examples of such an event. 3 Let A and B be two events with probabilities P (A) = 1/2,P (B) = 2/3,and P (A∩B) = 1/3. Define random variables X = 1 (A) and Y = 1 (B) ,where 1 (A) = 1 if Ais true and 0 otherwise. (a) Find the joint distribution of X and Y. (b) Obtain the conditional distribution of X given Y. Answer. (a) The joint distribution of X and Y is given by 1 P (X = 1,Y = 1) = P (A∩B) = , 3 1 1 1 P (X = 1,Y = 0) = P (A∩Bc) = P (A) −P (A∩B) = − = , 2 3 6 2 1 1 P