I am looking for a smooth probability density function with finite moments and closed form quantile function. As one knows, an example of smooth probability density function with finite moments is the Gaussian density funciton, but it does not have closed form quantile function. And an example of smooth probability density with closed form quantile function is the Cauchy density function, but it does not has finite moments. Finally, an example of probability density function with finite moments and closed form quantile function is derived by extending the exponential distribution density function by flipping it along the y axis and dividing it by 2, but the derived density function is not smooth at one point (the mean). I wonder whether one can construct a probability density function which is smooth, has finite moments and has closed-form quantile function.
Edit: where 'smooth' means infinitely differentiable and 'has finite moments' means all positive integer moments are finite