Lecture 11: The Poisson distribution

Sympathetic magic

• Don't confuse a r.v. with its distribution

  • Sum of r.v. vs Sum of PMF is different!

• "Word is not the thing, the map is not the territory."

  • random variable : random house, distribution : blue print(청사진)

Poisson distribution X ~ Pois(λ)

• $P(X=k) = e^{-\lambda} \lambda^{k} / k!$, $k \in \{0, 1, 2, ... \}$

  • $\lambda$ is a "rate(속도)" parameter(모수), $\lambda > 0$

• Valid : $\displaystyle \sum_{k=0}^{∞} \frac{e^{- \lambda} \lambda^{k}}{k!} = e^{- \lambda} e^{\lambda} = 1$ (Sum of prob. = 1)

  • cf. $e^{x} = \displaystyle \sum_{n=0}^{∞} \frac{x^n}{n!}$ (by Taylor series)

• $E(X) = e^{- \lambda} \displaystyle \sum_{k=0}^{∞} k \lambda^{k} / k!$

  $= e^{- \lambda} \displaystyle \sum_{k=1}^{∞} \frac{\lambda^{k}}{(k-1)!} = \lambda e^{- \lambda} \displaystyle \sum_{k=1}^{∞} \frac{\lambda^{k-1}}{(k-1)!}$

  $= \lambda e^{- \lambda} e^{\lambda} = \lambda$

• $V(X) = \lambda$, Variance is also $\lambda$!

• Often used for application where counting # of "successes" where there are a large # trials each with small prob. of success.

  • 여러 번의 시행을 하지만, 성공 가능성이 거의 없는 확률을 찾고자 할 때 쓰임

  (1) # emails in an hour.
  (2) # chips in chocolate chip cookie.
  (3) # earthquakes in a year in same region.

Pois Paradigm

• Events $A_1, A_2, ... , A_n$, $P(A_j) = p_j$, n large, $p_j$ is small

  • Events indep. or "weakly dependent" then # of $A_j$'s that occur is approximate

  • $Pois(\lambda)$, $\lambda = \displaystyle \sum_{j=1}^{n} p_j$

• $X \sim Bin(n, p)$, let n → ∞, p → 0, $\lambda = np$ is held constant.

  • Find what happens to $P(X=k) = \begin{pmatrix}n\\k\\ \end{pmatrix} p^{k} (1-p)^{n-k}$, k fixed.

  • $p = \lambda / n$,

    $P(X=k) = \frac{n(n-1)...(n-k+1)}{k!} (\frac{\lambda}{n})^k (1-\frac{\lambda}{n})^{n-k}$

    $= \frac{n(n-1)...(n-k+1) \lambda^{k}}{k! n^{k}} (1-\frac{\lambda}{n})^{n} (1-\frac{\lambda}{n})^{-k}$

    • n → ∞,

      $\frac{n(n-1)...(n-k+1) \lambda^{k}}{k!} → 1$, $(1-\frac{\lambda}{n})^{-k} → 1$

    • $(1+\frac{x}{n})^{n} → e^{x}$ as n → ∞

      : 복리 계산 식이라 생각하면 쉬움

    $P(X=k)$ → $\frac{\lambda^k}{k!} e^{- \lambda}$, Pois PMF at k.

  • 이항분포가 n → ∞, p → 0 조건을 가지면, 포아송 분포로 수렴함!

• ex.

  

  길바닥에 빗방울이 떨어지는 횟수 또한 포아송 근사로 설명할 수 있다.
  각 사각형에 빗방울이 떨어지는 사건은 이항분포이지만, 그 사건은 서로 독립이다.
  빗방울은 많이 떨어지지만 한 사각형 안에 떨어질 확률은 작기 때문에, 포아송 분포로도 볼 수 있다.

Birthday problem

• Ex. Have n people, find approximate prob. that there are 3 people with same birthday.

  • $\begin{pmatrix}n\\3\\ \end{pmatrix}$ triplets of people, indicator r.v. for each, $F_{ijk}$, $i < j < k$

• E(# triple matches) = $\begin{pmatrix}n\\3\\ \end{pmatrix} \frac{1}{365^2}$

  • 한 명의 생일은 정해져 있어야 하므로, 나머지 둘이 이와 같을 때의 확률로 계산!

• $X$ = # of triple matches. Approx Pois($\lambda$)

  • $\lambda = E(X) = \begin{pmatrix}n\\3\\ \end{pmatrix} \frac{1}{365^2}$

  • $I_{123} = I_{124}$ are not indep.

• $P(X \ge 1) = 1- P(X=0) \approx 1-e^{- \lambda} \lambda^0 / 0! = 1-e^{- \lambda}$

• 참고: Taylor Series, Poisson Distribution

• 출처 : Statistics 110, boostcourse