def normal_probability_between(lo, hi, mu=0, sigma=1):
    return Ch6.normal_cdf(hi, mu, sigma) - Ch6.normal_cdf(lo, mu, sigma)
def normal_lower_bound(probability, mu=0, sigma=1):
    """returns the z for which p(Z >= z) = probability"""
    return Ch6.inverse_normal_cdf(1 - probability, mu, sigma)
def normal_probabilty_above(lo, mu=0, sigma=1):
    return 1 - Ch6.normal_cdf(lo, mu, sigma)