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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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slide2
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

5 1 distribution function technique
5.1 Distribution Function Technique
  • ใช้ได้เฉพาะกรณี Continuous r.v.
  • พิจารณา Y = u (X1, X2, …, Xn)

1. หาค่า cdf of Y

2. จะได้ว่า

5 2 transformation technique one variable
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

slide5
Th’m1:ให้ f(x) เป็น pdf ของ continuous r.v. X, pdf ของ r.v.

Y = u(X) จะเป็น:

5 3 transformation technique two variables
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:

slide7
Th’m 2 (cont)

J is called Jacobian of the transformation and is the determinant

Elsewhere ,

5 4 moment generating function technique
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