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Chapter 8. Random Variable Transformation

Higher category: 【Statistics】 Statistics Overview

1. overview

2. moment generating function technique

3. distribution function technique

4. transformation technique

1. overview

⑴ random variable transformation: a methodology for obtaining p_Y(y) when Y = f(X) is present and p_X(x) is given 

⑵ Class 1. moment generating function technique

⑶ Class 2. distribution function technique: 2-step method

⑷ Class 3. transformation technique: 1-step method

⑸ Example problems for random variable transformation

⑹ Example problems for advanced random variable transformation

2. moment generating function technique

⑴ definition

⑵ moment generating function and probability distribution function are one-to-one correpondence

⑶ example: if X ~ N(μ, σ²), Y = aX + b ~ N(aμ + b, a²σ²)

3. distribution function technique 

⑴ definition

⑵ example 1. X ~ u[-1, 1], Y = X²

⑶ example 2. Y = max｛X₁, ···, X_n｝, X_i: i.i.d 

⑷ example 3. Y = min｛X₁, ···, X_n｝, X_i: i.i.d 

4. transformation technique

⑴ premise 

① it is only possible when the relationship between X and Y is one-to-one correspondence 

② by the premise, there exist two functions of Y = u(X) and X = ω(Y)

⑵ transformation technique of discrete random variable

⑶ transformation technique of continuous random variable 

① overview

○ when ω(Y) is a monotone increasing function, 

○ when ω(Y) is a monotone decreasing function, 

② generalization 

○ Jacobian: a kind of function determinant. geometrically, it means the area enlargement rate.

○ for x₁ = f₁^-1(y₁, y₂) and x₂ = f₂^-1(y₁, y₂),

○ if J ≠ 0, the one-to-one correspondence is established

③ tip. ways to get around a one-to-one correspondence.

○ premise: p_X(x) = e^-x (x ＞ 0), p_Y(y) = e^-y (y ＞ 0), Z = X + Y

○ question: p_Z(z)

○ calculation

Input : 2019.06.19 11:39