What is the simplest way to transform a normal$(mu,sigma)$ distribution to a beta$(a,b)$ distribution? I'm interested in knowing if there is an exact solution, but also if there are approximations that will achieve this? I'm not interested in a specific — ie normal$(3,2)$ to beta$(0.5,3)$ — transformation, but rather, general solutions.

**Contents**hide

#### Best Answer

If you're transforming from $Xsim N(mu,sigma^2)$ to $Ysim text{Beta}(alpha,beta)$, then $Y=F_Y^{-1}[Phi(frac{X-mu}{sigma})]$ where $F_Y$ is the desired cdf of $Y$ and $Phi$ is the standard normal cdf will achieve the result and preserve ordering (e.g. if $X_1<X_2$ and you transform $X_i$ to $Y_i$, for $i=1,2$ then $Y_1<Y_2$).

(I think this is effectively a duplicate but I didn't manage to locate the post I was looking for. If I find it this post will probably end up closing)

### Similar Posts:

- Solved – Translate exponential distribution into normal distribution
- Solved – Approximate the distribution of the sum of ind. Beta r.v
- Solved – expectation of $e^{-x}$ when x is log-normal
- Solved – expectation of $e^{-x}$ when x is log-normal
- Solved – Compute truncated normal distribution with specific mean and variance