Chapter 3.8 Quiz ENGR200

True/False — Poisson distribution concepts.

1) Poisson\((\lambda)\) models the number of events in a fixed interval when events occur independently at a constant average rate \(\lambda\).

2) The PMF of Poisson\((\lambda)\) is \(P(X=k)=e^{-\lambda}\,\lambda^{k}/k!\) for \(k=0,1,2,\dots\)

3) For Poisson\((\lambda)\), the mean is \(\lambda\) and the variance is also \(\lambda\).

4) Poisson is memoryless: \(P(X>m+n\mid X>m)=P(X>n)\).

5) If the observed rate is \(r\) events per minute, then the expected count in a \(t\)-minute window is Poisson with parameter \(\lambda = r/t\).

6) For Poisson\((\lambda)\), \(P(X\ge 1)=e^{-\lambda}\).

7) The sum of independent Poisson\((\lambda_{1})\) and Poisson\((\lambda_{2})\) is Poisson\((\lambda_{1}+\lambda_{2})\).

8) For large \(n\) and small \(p\) with \(np=\lambda\) fixed, \(\mathrm{Binomial}(n,p)\) is well approximated by \(\mathrm{Poisson}(\lambda)\).

9) In Poisson\((\lambda)\), it’s possible that \(P(X = 2.5) > 0\).

10) If the event rate doubles, the expected count doubles but the variance quadruples for Poisson.

Score: 0/10 correct