and now(9), with X*=X, follows for a large class of functions f by inte- gration by parts We can gain some additional intuition regarding the zero bias transfor mation by observing its action on non-normal distributions, which, in some sense, moves them closer to normality. Let b be a bernoulli random variable with success probability P E(0, 1), and let ua, b denote the uniform distri bution on the finite interval [a, b]. Centering B to form the mean zero discrete random variable X=B-p having variance o=p(1-P), substitution into he right hand side of (9) yields EIXf(XJ= E(B-P)f(B-p p(1-p)f(1-p)-(1-p)pf(-p) 2[f(1-p)-f(-p f(u) f(U), for U having uniform density over [-p, 1-Pl. Hence, with=d indicating the equality of two random variables in distribution (B-p=dU where U has distribution Z-p, 1-pI This example highlights the general fact that the distribution of X* is always absolutely continuous, regardless of the nature of the distribution of X It is the uniqueness of the fixed point of the zero bias transformation, that is, the fact that X* has the same distribution as X only when X is normal that provides the probabilistic reason behind the Clt. This only if'direction of Steins characterization suggests that a distribution which gets mapped to one nearby is close to being a fixed point of the zero bias transformation, and therefore must be close to the transformation's only fixed point, the normal Hence the normal approximation should apply whenever the distribution of a random variable is close to that of its zero bias transformation Moreover, the zero bias transformation has a special property that imme- diately shows why the distribution of a sum Wn of comparably sized inde- pendent random variables is close to that of Wr: a sum of independent terms can be zero biased by replacing a single summand chosen proportionally to its variance and replacing it with one of comparable size. Thus, by differ ing only in a single summand, the variables Wn and W are close, makingand now (9), with X∗ = X, follows for a large class of functions f by inte￾gration by parts. We can gain some additional intuition regarding the zero bias transfor￾mation by observing its action on non-normal distributions, which, in some sense, moves them closer to normality. Let B be a Bernoulli random variable with success probability p ∈ (0, 1), and let U[a, b] denote the uniform distri￾bution on the finite interval [a, b]. Centering B to form the mean zero discrete random variable X = B − p having variance σ 2 = p(1 − p), substitution into the right hand side of (9) yields E[Xf(X)] = E[(B − p)f(B − p)] = p(1 − p)f(1 − p) − (1 − p)pf(−p) = σ 2 [f(1 − p) − f(−p)] = σ 2 Z 1−p −p f 0 (u)du = σ 2Ef0 (U), for U having uniform density over [−p, 1 − p]. Hence, with =d indicating the equality of two random variables in distribution, (B − p) ∗ =d U where U has distribution U[−p, 1 − p]. (10) This example highlights the general fact that the distribution of X∗ is always absolutely continuous, regardless of the nature of the distribution of X. It is the uniqueness of the fixed point of the zero bias transformation, that is, the fact that X∗ has the same distribution as X only when X is normal, that provides the probabilistic reason behind the CLT. This ‘only if’ direction of Stein’s characterization suggests that a distribution which gets mapped to one nearby is close to being a fixed point of the zero bias transformation, and therefore must be close to the transformation’s only fixed point, the normal. Hence the normal approximation should apply whenever the distribution of a random variable is close to that of its zero bias transformation. Moreover, the zero bias transformation has a special property that imme￾diately shows why the distribution of a sum Wn of comparably sized inde￾pendent random variables is close to that of W∗ n : a sum of independent terms can be zero biased by replacing a single summand chosen proportionally to its variance and replacing it with one of comparable size. Thus, by differ￾ing only in a single summand, the variables Wn and W∗ n are close, making 6