tailieunhanh - A Course in Mathematical Statistics phần 3

Tham khảo tài liệu 'a course in mathematical statistics phần 3', ngoại ngữ, ngữ pháp tiếng anh phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 90 4 Distribution Functions Probability Densities and Their Relationship Refer to Exercise in Chapter 3 and determine the . s corresponding to the . s given there. Refer to Exercise in Chapter 3 and determine the . s corresponding to the . s given there. Let X be an . with . F. Determine the . of the following . s -X X2 aX b XIab X when i X is continuous and F is strictly increasing ii X is discrete. Refer to the proof of Theorem 1 iv and show that we may assume that xn ị -rc xn T rc instead of xn - xn rc . Let f and F be the . and the . respectively of an . X. Then show that F is continuous and dF x dx f x at the continuity points x of f. i Show that the following function F is a . Logistic distribution and derive the corresponding . f. Fx . x efi a 0 p efi ii Show that f x aF x 1 - F x . Refer to Exercise in Chapter 3 and determine the . F corresponding to the . f given there. Write out the expressions of F and f for n 2 and n 3. If X is an . distributed as N 3 use Table 3 in Appendix III in order to compute the following probabilities i P X -1 ii P X iii P X . The distribution of IQ s of the people in a given group is well approximated by the Normal distribution with J 105 and 5 20. What proportion of the individuals in the group in question has an IQ i At least 150 ii At most 80 iii Between 95 and 125 A certain manufacturing process produces light bulbs whose life length in hours is an . X distributed as N 2 000 2002 . A light bulb is supposed to be defective if its lifetime is less than 1 800. If 25 light bulbs are The . of a Random Vector and Its Properties 91 tested what is the probability that at most 15 of them are defective Use the required independence. A manufacturing process produces 1 -inch ball bearings which are assumed to be satisfactory if their diameter lies in the .