Random Variable

독립적인(independent) 베르누이 시행에서 첫 번째 성공이 일어나기까지의 시행횟수

분포의 특성

• pmf of $X(Y_i: Bernoulli \space r.v.)$
  $P(X=1)=P(Y_1=1)=p,\\ P(X=2)=P(Y_1=0,Y_2=1)=P(Y_1=0)P(Y_2=1)=(1-p)p,\\ P(X=3)=P(Y_1=0,Y_2=0,Y_3=1)=(1-p)^2p,\\ \vdots\\ P(X=x)=(1-p)^{x-1}p \space (x=1,2,...)$
  
  
• 기댓값
  $E(X)=\frac{1}{p}$
  pf)
  $E(X)=\sum_{x=1}^{\infty}x(1-p)^{x-1}p=p(1+2(1-p)+3(1-p)^2+...),\\ (1-p)E(X)=p((1-p)+2(1-p)^2+3(1-p)^3+...),\\ E(X)-(1-p)E(X)=p(1+(1-p)+(1-p)^2+(1-p)^3+...)=pE(X),\\ E(X)=1+(1-p)+(1-p)^2+...=\frac{1}{1-(1-p)}=\frac{1}{p}$
  
  
• 분산
  $Var(X)=\frac{1-p}{p^2}$
  pf)
  
  
  
• mgf
  $M_X(t) = \sum_{x=1}^{\infty}e^{tx}(1-p)^{x-1}p=p \cdot e^t\sum_{x=1}^{\infty}(e^t(1-p))^{x-1}=\frac{p \cdot e^t}{1-e^t(1-p)} \space (e^t(1-p)<1)$

• 참고자료
  - https://blog.naver.com/PostView.naver?blogId=chunsa0127&logNo=222049190534&photoView=3