Order Statistics for IIND Variables

Question

Given 3 exponential random variables with different means (for example 1, 2, 3), how can one calculate E(X) for MIN(X1,X2,X3)?

Answer 1

Let $X_{(1)}$ be the minimum of $X_1,X_2$ and $X_3$. Then $P(X_{(1)}>x)=P(X_1>x,X_2>x,X_3>x)=\prod_1^3 P(X_i>x)$.

Then use the fact that $E(X_{(1)})=\int_0^{\infty}P(X_{(1)}>t)dt$ or you can differentiate $1-P(X_{(1)}>x)$ w.r.t. $x$ to find the density at $x$

Answer 2

Presumably you wish to calculate $\mathsf E(\min(X_1,X_2,X_3))$

Then where $f_n(x) = \lambda_n e^{-\lambda_n x}, F_n(x)=1-e^{-\lambda_n x}$ are the probability density function and cumulative distribution functions (respectively) for the exponentially distributed random variables $\{X_n\}_3$.

$\begin{align}\mathsf E(\min(X_1,X_2,X_3)) = & { \int_0^\infty x_1 f_1(x_1)\int_{x_1}^\infty f_2(x_2)\int_{x_1}^\infty f_3(x_3)\operatorname d x_3\operatorname d x_2\operatorname d x_1 \\ + \int_0^\infty x_2 f_2(x_2)\int_{x_2}^\infty f_1(x_1)\int_{x_2}^\infty f_3(x_3)\operatorname d x_3\operatorname d x_1\operatorname d x_2 \\ + \int_0^\infty x_3 f_3(x_3)\int_{x_3}^\infty f_1(x_1)\int_{x_3}^\infty f_2(x_2)\operatorname d x_2\operatorname d x_1\operatorname d x_3 } \\[1ex] = & { \int_0^\infty x f_1(x) (1-F_2(x))(1-F_3(x)) \operatorname d x \\ + \int_0^\infty x f_2(x) (1-F_1(x))(1-F_2(x)) \operatorname d x \\ + \int_0^\infty x f_3(x) (1-F_2(x))(1-F_1(x)) \operatorname d x } \\[1ex] = & \ldots \end{align}$

Which leads to a nice result. Can you complete?

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Alternatively: Consider that Exponential random variables describe the time until the next event in a Poisson process. That is where the events occur independently (of other events in their own process) and at constant average rates $(\lambda_1, \lambda_2, \lambda_3)$.

So, if the arrivals of events of the three types are also independent of each other, the arrival of the next event of any type will also be a Poisson process, and the time until the next event of any type will be exponentially distributed.   The expected time of this is the expected minimum of the times for each type.   That is it is what we seek.

Can you determine the rate of arrival of the next event of any type, in terms of the three rates?