Minimum of two random variables with exponential distribution

Question

Let $X,Y$ be two random variables with exponential distribution and their rates are $\gamma, \beta $. Let $Z$ be a random variable such that $Z = min\{X, Y\}$.

How do I prove that the density function of $Z$ is $(\gamma+\beta)e^{-(\gamma+\beta)x}$ for $x\geq0$ and $0$ for $x<0$?

Answer 1

HINT

Following the hint in the comment, note that $$ \begin{split} F_Z(z) &= \mathbb{P}[Z \le z] = \mathbb{P}[\min\{X,Y\} \le z] \\ &= \mathbb{P}[X \le z, Y \le z] \quad \text{by independence of $X,Y$}\\ &= \mathbb{P}[X \le z] \cdot \mathbb{P}[Y \le z] \\ &= F_X(z) \cdot F_Y(z) \end{split} $$ And now you can compute $$ f_Z(z) = \frac{d}{dz}\left[F_Z(z)\right]. $$