4. Distribution of Functions of Random Variables

• 우리는 왜 functions of random variables의 분포를 알고싶어할까?
  • 우리는 모집단의 특성을 설명해줄 ‘모수’를 추론하기 위해, 표본으로부터 모수를 추정하고 그 추정량의 정확도를 검정한다. 추정량을 검정하기 위해 \(P(|\hat{\theta}-\theta| \leq c)\)를 구하기 위해서는 \(\hat{\theta}\) = statistic = \(T(X_1, X_2, \cdots, X_n)\)= functions of random variables 의 분포를 알아야 한다.
• Methods to obtain the distribution of functions of random variables
  • distribution function technique
    • continuous type random variables \(X_1, \cdots, X_n \sim\) pdf \(f(x_1, \cdots, x_n)\)→ What is pdf of \(Y=u(X_1, \cdots, X_n)\)?
      1. compute cdf of \(Y\): \(F(y)=P[u(X_1, \cdots, X_n) \le y]\)
      2. obtain pdf of \(Y\) by differentiate \(F(y)\): \(f(y) = \frac{d}{dy}f(y)\)
  • transformation technique
    • discrete random variable: \(p_Y(y)=p_X(u^{-1}(y))\)
    • continuous random variable: \(f_Y(y) = f_X(g^{-1}(y))|\frac{dx}{dy}|\)
  • moment generating function technique
    • \(\because\) mgf is unique and completely determines the distribution of the random variable
  • asymptotic technique
    • definition of limiting distribution
    • mgf technique
    • Central limit theorem + Slutsky’s theorem or Delta method
  • Monte Carlo Method