Random Variable

단위 시간(또는 단위 공간) 내에 사건이 발생하는 횟수

분포의 특성

• pmf of $X$
  $P_X(x)=\frac{e^{-\lambda} \lambda^x}{x!} \space (x=0,1,2,...)(\lambda>0)$
  
  $\sum_{x=0}^{\infty}\frac{e^{-\lambda} \lambda^x}{x!}=1?\\ From \space Talyor \space expansion, e^t=\sum_{x=0}^{\infty}\frac{t^x}{x!}\\ \sum_{x=0}^{\infty}\frac{e^{-t}t^x}{x!}=1 \space for \space any \space t$
  
  
• 기댓값
  $E(X)=\lambda$
  pf)
  $E(X)=\sum_{x=0}^{\infty}x\frac{e^{-\lambda} \lambda^x}{x!}=\sum_{x=1}^{\infty}\frac{e^{-\lambda} \lambda^x}{(x-1)!}\\ Let \space t=x-1\\ \sum_{t=0}^{\infty}\frac{e^{-\lambda} \lambda^{t+1}}{t!}=\sum_{t=0}^{\infty}\lambda\frac{e^{-\lambda} \lambda^t}{t!}=\lambda$
  
  
• 분산
  $Var(X)=\lambda$
  pf)
  $E(X(X-1))=\sum_{x=0}^{\infty}x(x-1)\frac{e^{-\lambda} \lambda^x}{x!}=\sum_{x=2}^{\infty}\frac{e^{-\lambda} \lambda^x}{(x-2)!}\\ Let \space t=x-2\\ \sum_{t=0}^{\infty}\frac{e^{-\lambda} \lambda^{t+2}}{t!}=\sum_{t=0}^{\infty}\lambda^2\frac{e^{-\lambda} \lambda^t}{t!}=\lambda^2=E(X^2)-E(X)\\ E(X^2)=\lambda^2+\lambda\\ Var(X)=E(X^2)-[E(X)]^2=\lambda^2+\lambda-\lambda^2=\lambda$
  
  
• mgf
  $M_X(t)=\sum_{x=0}^{\infty}e^{tx}\frac{e^{-\lambda} \lambda^x}{x!}=\sum_{x=0}^{\infty}\frac{e^{-\lambda} (e^t\lambda)^x}{x!}=e^{-\lambda}\sum_{x=0}^{\infty}\frac{(e^t\lambda)^x}{x!}=e^{-\lambda}e^{\lambda e^t}=e^{\lambda(e^t-1)} \space (-\infty<t<\infty)$