I'm learning about exponential distributions, and we've been given this distribution $Z = \min\left[X, Y\right]$, where $X$ and $Y$ follow exponential distributions.
How can I visualize this?
If $X \sim \operatorname{Exp}(2)$, and $Y \sim\operatorname{Exp}(3)$, then $f_X(x) = 2e^{-2x}$ and $f_Y(x) = 3e^{-3x}$.
Would that just mean that $f_Z(x) = \min\left[2e^{-2x}, 3e^{-3x}\right]$, i.e. a piecewise function containing both distributions?
And how is $Z$ another exponential distribution if it is piecewise? I believe I am misunderstanding the meaning of minimum.
The output of the minimum function will give the smallest of an observation of $X$ and $Y$ but not the p.d.f. itself since a value of a random variable does not take the value of the p.d.f. but rather take the "probability" from the p.d.f.
In general the trick is that if $Z>z$ and $Z=\min\left[X,Y\right]$, then it implies that $X>z$ and $Y>z$. Assuming $X$ and $Y$ are independent, you will have
$$P(Z>z)=P(X>z)\cdot P(Y>z)$$
When you work everything out, you will have $Z\sim\operatorname{Exp}(\lambda_x+\lambda_y)$.