A truncated discrete distribution is one in which a particul

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

P(X(T) =x)= P(X =x)/P(X > 0), x=1,2,...

Find the pmf, mean and variance of the 0-truncated random variable starting from

(a) X ~ Poisson().

(b) X ~ negative binomial(r, p).

Solution