Consider $K$ independent Laplace variables $X_i$ ($1 \leq i \leq K$) with mean 0 and scale $\lambda$. Let $X′$ be the variable taking the value of the Laplace variable whose absolute value is the minimum among all $X_i$'s. Due to the randomness of $X_i$'s, $X′$ may not always equal a particular $X_i$. I would like to know what the CDF of $X′$ is. Does it also follow Laplace distribution? How to prove or disprove that? Many thanks!

Consider $K$ independent Laplace variables $X_i$ ($1 \leq i \leq K$) with mean 0 and scale $\lambda$. Let $X′$ be the variable taking the value of the Laplace variable whose absolute value is the minimum among all $X_i$'s. Due to the randomness of $X_i$'s, $X′$ may not always equal a fixed $X_i$. I would like to know what the CDF of $X′$ is. Does it also follow Laplace distribution? How to prove or disprove that? Many thanks!