Question
Suppose $X$ is a Poisson random variable with $E[X]=\alpha$.(a) Prove that, if $P[X=k]=P[X=k+1]$, then $\alpha=k+1$.(b) Prove that, if $\alpha=k+1$, then $P[X=k]=P[X=k+1]$.
Step 1
The probability that $X$ equals $k$ is given by: \[ P[X = k] = e^{-\alpha} \frac{\alpha^k}{k!} \] for $k = 0, 1, 2, \dots$. Show more…
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