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 0-truncated random variable XT has probability mass function

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

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

• X ~ Poisson ( )
• X ~ negative binomial (r, p) as in P(X = x) = (r+x1x)pr(1p)x,x=0,1,...