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$X_1$, $X_2$, $X_3$ 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.

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Hint: We have $Y\gt y$ if and only if all the $X_i$ are $\gt y$.

From this you should be able to quickly find the cumulative distribution function of $Y$, and then, if you wish, the density.

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First: to find distribution of $Y$: $$\Pr(Y \leq y) = 1-\Pr(\min(X_1, X_2, X_3)>y)=1- \Pr(X_1>y, X_2>y, X_3>y)=$$ $$=1-\prod_{i=1}^3 \Pr(X_i >y)$$ If $F_X(y)$ and $F_Y(y)$ are distributions of $X$ and $Y$ respectively then: $$ F_Y(y)=1- \prod_{i=1}^3(1-F_X(y))$$ Second: You know distribution of $X_i$ (or you can with ease derive). If you need the pdf just differentiate.

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  • $\begingroup$ I don't get how you got the distribution of $Y$, $F_Y$(y), in the third line. Also, do you do you mean the pdf of the exponential function in the second part? $\endgroup$ – Sue Apr 10 '13 at 23:27
  • $\begingroup$ Do you know about the relationship between $\Pr(Y \leq y)$ and cumulative distribution function $F_Y(y)$? If not you probably need a statistical textbook. pdf stands for a probability density function. $\endgroup$ – Rob Apr 11 '13 at 8:39
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The wiki on exponential distribution has an answer to that. The answer of course is exponential distribution. http://en.wikipedia.org/wiki/Exponential_distribution#Distribution_of_the_minimum_of_exponential_random_variables

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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?

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