What is the official name of this distribution, $$k\exp(-\lambda|x|^\alpha),$$ where $\alpha$ is usually less than 1, but greater than 0? More importantly, could anyone tell me what the variance of this distribution is? Thank you very much.

Thank you all.

This distribution can be called generalized normal distribution, whose standard form is

$$f(x)=\beta/2\alpha\Gamma(1/\beta)\exp(-|(x-\mu)/\alpha|^\beta)$$

whose variance is $$\alpha^2 \Gamma(3/\beta)/\Gamma(1/\beta)$$