# Conditional Expectation for Probability Distribution [duplicate]

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How is the minimum of a set of random variables distributed?

I have two RVs from the same distribution (exponential distribution with parameter λ). How do I calculate E(min|min>x)? I know how to get E(X|X>x) but I specifically need X to be the smallest of the two RVs.

• If you have the density function for the minimum, then it's exactly the same as doing the calculation for an arbitrary random variable, $X$. Are you asking how to derive the density of the minimum? Apr 12 '12 at 22:39
• Sorry for confusion - yes, that is basically what I am asking. Apr 12 '12 at 22:40
• This is a commonly asked question on here. See, for example, this thread: stats.stackexchange.com/questions/220/… Apr 12 '12 at 22:43

An exponential distribution is the time for the first occurrence of a Poisson process with rate $\lambda$ so the minimum of $n$ iid exponentially distributed random variables is the time for the first occurrence of a Poisson process with rate $n\lambda$ and so is exponentially distributed with mean $\frac{1}{n\lambda}$.
But an exponential distribution is memoryless, so $E(\min | \min > x) = x+ \frac{1}{n\lambda}$.
In your particular question, $n=2$.