Minimum and Race of Independent Exponential Clocks

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Exponential Distribution / Memoryless Property.

Your Task
Let $X_1, X_2, \dots, X_n$ be independent exponential random variables with rates $\lambda_1, \lambda_2, \dots, \lambda_n > 0$. Let $M = \min(X_1, \dots, X_n)$.
(1) Prove that $M$ is exponentially distributed with rate $\Lambda = \sum_{i=1}^n \lambda_i$.
(2) Prove that the probability that $X_k$ is the minimum is $\frac{\lambda_k}{\sum_{i=1}^n \lambda_i}$, and that the index of the minimum is independent of the value of the minimum.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.