Study of Moment Generating Functions in Probability and Statistics
Explore the concepts of moment generating functions in probability and statistics, including properties and calculations of mean and variance. Learn about factorial moment generating functions and characteristic functions, with detailed theorems and examples discussed in the lecture.
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Probability and Statistics Third Class By Dr. Jawad Mahmoud Jassim Dep. of Math. College of Education for Pure Sciences University of Basrah Iraq Add a footer 1
In the pervious lecture we study the relation between the moments generating function, and how to calculate the mean and the variance by using the moment generating function. In this lecture we take some properties of the moment generating function. Also, we take the factorial moment generating function and the characteristic function. 2 and the moment Add a footer 2
Theorem: If ?(?) be the moment generating function of the r.v. ? such that ? ? = ?0+ ?1t + ?2?2+ + ????+ , then ? ??= (n!)?? Example(1): Let ? be a r.v. m.g.f. ? ? = the 100th moment of ? ? 1 1+? . What is 3 Add a footer 3
Solution: Since ? ? = Here, ??= ( 1)? . Then by above theorem we get ? ?100= 100 ! . 1 1+?= 1 ? + ?2 ?3+ ?4 + 1???+ Theorem: Let ? be a r.v. with m.g.f. ??t . Let ? and ? are any two real numbers, then (1) ??+?t = ?????t (2) ???t = ??bt (3) ???+?t = ?????(bt) Add a footer 4
Example(2): If the r.v. ? has m.g.f. ? ? = ? = 1 2? . What is the m.g.f. of ? ? Solution: By above theorem; here ? = 2 ??? ? = 1. Hence, the m.g.f. of ? is ??t = 1 2? for ? <1 1 2 . If ?? 1+4? . Add a footer 5
Example(3): Let ? be a r.v. with p.d.f. ? ? = 6? 1 ? ,0 < ? < 1 0 Let ? = 3? + 2. Find ? ? ,??? ? ??? ??t . Solution: ? ? = ? 3? + 2 = 3? ? + 2 = 3 0 = 18 0 ??? ? = ? ?2 (? ? )2 . Now,? ?2= E 9?2 12X + 4 = 9E ?2 12E X + 4 Add a footer ,?? ?????? 1?? ? ?? + 2 1(?2 ?3)dx + 2 =1 2 . 6
1 1 ? ?2= 9 ?2? ? ?? 12 ?? ? ?? + 4 0 0 7 10 . 7 10 1 ??? ? = ??= E ??? 1 = 9 20 . 4= ???? ? ?? =6??? 12?? 6? + 12 = ,t 0 ?3 0 Add a footer 7
Therefore, the m.g.f of ? is ??t = ?2? 18?? 3? 12? 3?+ 12 27?3 =6?? ?+ 4? ? 4?2? 9?3 ,t 0 . Definition: The factorial moment generating function (f.m.g.f.) of a r.v.? , denoted by ?(?), is defined as: ? ? = ?(??) Add a footer 8
Question: What is the relation between the moment generating function and the factorial moment generating function ? Answer: The relation is given by: ? ? = ?(???). Proof: ? ? = ? ??= E ????? = E ?????= M(lnt) . Add a footer 9
Example(4): If the m.g.f. of the r.v. ? is ? ? = f.m.g.f. of ? ? Solution: ? ? = M lnt = Definition: The characteristic function (?) of the r.v. ? is defined as: ? = ?(????),? = ?? 1+4? . What is the ? 1+4??? . 1 . Add a footer 10
Summary In this lecture we take some properties of the moment generating function: ??+?t = ?????t ???t = ??bt ???+?t = ?????(bt) Also, we take the factorial moment generating function and its relation with the moment generating function, which is : ? ? = ?(???). Add a footer 11
Assignment: Choose the correct answer: (1) If ? ? = ?( (a) 1 (b) 2 (c) 3 (d) 4 (2) If the f.m.g.f. of ? is ? ? = (a) zero (b) -3 (c) 1 (d) Non of these. (3) If the m.g.f. of r.v. ? is: ? ? = (1 2?) 3,then the m.g.f.of r.v. ? = 3? + 5 is: (a) ? 3?(1 10?) 3 (b) ? 3?(1 2?) 3 (c) ?5?(1 2?) 3 (d) ?5?(1 + 6?) 3. (4) If ? ? = ?2?+2?2= 1, then ? = (a) 1 (b) -1 (c) 0.5 (d) Non of these . ?? 4+2?) is the m.g.f. of r.v. ? ,then ? = ? 1+4??? , then the mean of ? is: Add a footer 12
