Distribution of a random variable 
$X_1$, $X_2$, $X_3$ are independent random variables, each with an exponential distribution, but with means of $2.0, 5.0, 10.0$ respectively. Let $Y$= the smallest or minimum value of these three random variables. Derive and identify the distribution of $Y$. (The distribution function may be useful). 

How do I solve this question? Do I plug in each mean to the exponential distribution? I would appreciate it if someone could explain this to me, thanks.

A "well-known" property of the exponential distribution (possibly mentioned in your textbook or notes) is that the minimum of independent exponentials with rates $r_1, \ldots, r_n$ (i.e. means $1/r_1, \ldots, 1/r_n$) is exponential with rate $r_1 + \ldots + r_n$ (mean $1/(r_1 + \ldots + r_n)$).  But from the hint it sounds like you're expected to calculate the cumulative distribution function.  Note that 
$\min(X_1, \ldots, X_n) > t$ if and only if all $X_i > t$.  Can you find the 
probability of that?
