Continuous Distributions without a Closed-Form Inverse

• Many continuous distributions don't have a closed-form inverse function. Strictly speaking, the inverse transform method discussed earlier cannot be applied.

• If we are willing to accept numeric solution, inverse functions can be found. Here we have an example for normal distribution.

• One of the inverse cdf of the standard normal distribution was proposed by Schmeiser:
  
  $$X = F^{-1}(R) \approx \frac{R^{0.135} - (1 - R)^{0.135}} {0.1975}$$
  
  
  for $0.0013499 \le R \le 0.9986501$ which matches the true normal distribution with one digit after decimal point. See Table 9.6 on page 335.