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.