This is a basic question that I can't answer (it's not a homework assignment).  A reference would be perfectly acceptable.Consider a $[0,k]$ valued random variable X distributed binomially so that:$$\text{Pr}[X=i]=\binom{k}{i}/2^k$$It's clear that we can sample from $X$ in poly($k$) time using $k$ random bits.  The basic question is if it's possible to do better if I'm only interested in approximate sampling. For example, if I'm interested in sampling from any $Y$ so that $|X-Y|<\epsilon$ (in statistical distance), can I do this in randomized time poly($1/\epsilon$) (i.e., with $1/\epsilon$ random bits)?
Sampling from the binomial distribution with limited randomness and time
probabilistic computation