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Chapter 5 : Functions of Random Variables

Chapter 5 : Functions of Random Variables. สมมติว่าเรารู้ joint pdf ของ X 1 , X 2 , …, X n --> ให้หา pdf ของ Y = u (X 1 , X 2 , …, X n ) 3 วิธี 1. Distribution Fn Technique 2. Transformation Technique 3. MGF Technique. 5.1 Distribution Function Technique.

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Chapter 5 : Functions of Random Variables

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  1. Chapter 5: Functions of Random Variables

  2. สมมติว่าเรารู้ joint pdf ของ X1, X2, …, Xn --> ให้หา pdf ของ Y = u (X1, X2, …, Xn) • 3 วิธี 1. Distribution Fn Technique 2. Transformation Technique 3. MGF Technique

  3. 5.1 Distribution Function Technique • ใช้ได้เฉพาะกรณี Continuous r.v. • พิจารณา Y = u (X1, X2, …, Xn) 1. หาค่า cdf of Y 2. จะได้ว่า

  4. 5.2 Transformation Technique: One Variable กรณี Discrete random variable กรณี Discrete random variable กรณี Continuous random variable กรณี Discrete random variable กรณี Continuous random variable • Assume that the function y = u(x) is “differentiable” and “monotonic” (either increasing or decreasing) for all X in which f(x) 0 --> Inverse fn x = w(y) exists and is differentiable for all corresponding values of y

  5. Th’m1:ให้ f(x) เป็น pdf ของ continuous r.v. X, pdf ของ r.v. Y = u(X) จะเป็น:

  6. 5.3 Transformation Technique : Two Variables • Th’m 2Let be the value of joint pdf of cont rv X1 and X2 at . If fn given by and are partially differentiable with X1 and X2and represent a one-to-one transformation for all values within the range of X1 and X2 for which , then, for these values of x1and x2 , the equations and can be uniquely solved for x1 and x2 to give and and for the corresponding values of and the joint pdf of and is given by:

  7. Th’m 2 (cont) J is called Jacobian of the transformation and is the determinant Elsewhere ,

  8. 5.4 Moment-Generating Function Technique • Th’m 3ถ้า เป็นindependent rvและ ดังนั้น คือ ค่าของ moment-generating function ของ Xiat t • ถ้า MY(t) = MZ(t) จะได้ว่า Y กับ Z มี pdf เหมือนกัน

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