Random Variable

독립적인(independent) 베르누이 시행에서 r 번 성공하기 전까지의 실패 횟수. 즉, 기하분포의 일반화

분포의 특성

• pmf of $X(Y_i: Bernoulli \space r.v.)$
  $P(X=0)=P(Y_1=1,Y_2=1,...,Y_r=1)=p^r \\ P(X=1)=\binom{r}{r-1}p^{r-1}(1-p)p=\binom{r}{r-1}p^{r}(1-p) \\ \vdots\\ P(X=x)=\binom{r+x-1}{r-1}p^{r-1}(1-p)^xp=\binom{r+y-1}{r-1}p^r(1-p)^x \space (x=0,1,2,...)(0<p<1)$
  
  
• 기댓값
  $E(X)=\frac{r(1-p)}{p}$
  pf)
  $E(X)=\sum_{x=0}^{\infty}x\binom{r+x-1}{r-1}p^r(1-p)^x=\sum_{x=1}^{\infty}\frac{(r+x-1)!x}{(r-1)!x!}p^r(1-p)^x=\sum_{x=1}^{\infty}\frac{(r+x-1)(r+x-2)!}{(r-1)!(x-1)!}p^r(1-p)^x\\ Let \space z=x-1\\ \sum_{z=0}^{\infty}\frac{(r+z-1)!}{(r-1)!z!}p^r(1-p)^z(1-p)(r+z)=(1-p)r+(1-p)E(Z)\\ E(X)-(1-p)E(X)=(1-p)r \space(b/c \space E(X)=E(Z))\\ E(X)=\frac{r(1-p)}{p}$
  
  
• 분산
  $Var(X)=\frac{r(1-p)}{p^2}$
  pf)
  
  
• mgf
  $M_X(t)=\sum_{x=0}^{\infty}e^{tx}\binom{r+x-1}{r-1}p^r(1-p)^x=\sum_{x=0}^{\infty}\binom{r+x-1}{x}p^r[e^t(1-p)]^x=p^r(1-(1-p)e^t)^{-r} \space by \space 음이항정리 \space (t<-log(1-p))$