1STAT 702/J702 B.Habing Univ. of S.C. 1STAT 702/J702 September 30th, 2004-Lecture 13-Instructor: Brian HabingDepartment of StatisticsTelephone: 803-777-3578E-mail: [email protected] 702/J702 B.Habing Univ. of S.C. 2Today• Functions of Continuous Distributions (continued)• Joint DistributionsSTAT 702/J702 B.Habing Univ. of S.C. 32.2.2-The Gamma Distribution2.2.3 – The Normal Distribution0 )()(1≥Γ=−−tettgtforλαααλ∞<<∞−=−−xexfx for 222)(21)(σµπσ2STAT 702/J702 B.Habing Univ. of S.C. 42.3 - Functions of Random VariablesLet Y=a X+bFY(y ) = FX( (y –b)/a ) f Y(y ) =(1/a ) f X( (y –b)/a )e.g. if X~N(µ, σ 2) then (X-µ)/ σ∼ΖSTAT 702/J702 B.Habing Univ. of S.C. 5Y=g (X): Let X be a continuous RV with p.d.f. f (x ) and Y=g (X), where g is differentiable and strictly monotone every where that f (x)>0.Then)())(()(11ygdydygfyfXY−−=STAT 702/J702 B.Habing Univ. of S.C. 6Page 60: “For any specific problem, it is usually easier to proceed from scratch than to decipher the notation and apply the proposition.”3STAT 702/J702 B.Habing Univ. of S.C. 7Example) Let X=Z2where Z~N(0,1)STAT 702/J702 B.Habing Univ. of S.C. 8Example 2) Let X=F-1(U) where U is uniform on [0,1] and F is a CDF.STAT 702/J702 B.Habing Univ. of S.C. 9Chapter 3 – Joint DistributionsThe joint behavior of two random variables X and Y is determined by there CDF:FXY(x ,y) = P(X≤x, Y≤y)4STAT 702/J702 B.Habing Univ. of S.C. 10We can use this definition to find the area of any given rectangle:P(x1< X≤ x2, y1< Y≤ y2) = FXY(x2, y2) – FXY(x1, y2) –FXY(x2, y1) + FXY(x1, y1), for x1<x2, y1<y2. STAT 702/J702 B.Habing Univ. of S.C. 113.2 - Discrete R.V.’sFor discrete R.V.’s the joint p.m.f. is p(x, y ) = P(X=x, Y=y )STAT 702/J702 B.Habing Univ. of S.C. 12Example) A fair coin is tossed three times. Let X=number of heads in three tossings and Y= difference (in absolute values) between the number of heads and number of