A truncated discrete distribution is one in which a particular class cannot be observed and is eliminated from the sample space. In particular, if X has range 0, 1, 2,... and the 0 class cannot be observed (as is usually the case), the O-truncated random variable XT has pmf

Find the pmf, mean, and variance of the O-truncated random variable starting from
(a) X ~ Poisson(λ).
(b) X ~ negative binomial(r, p), as in (3.2.10).

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P(X=z)