Question

What is the parameter of exponential distribution for min(x,y)?

Answer 1

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$\mathfrak{\blue{Answer.}}$Note that you must assume that XX and YY are independent, otherwise the result is easily seen to be false.

There is a constant λλ such thatP(X≥t)=e−λtP(X≥t)=e−λt for every t>0t>0.

There is a constant μμ such thatP(Y≥t)=e−μtP(Y≥t)=e−μt for every t>0t>0.

Then for every t>0t>0 we have

P(Z≥t)=P(X≥t,Y≥t)=P(X≥t)P(Y≥t)=e−(λ+μ)tP(Z≥t)=P(X≥t,Y≥t)=P(X≥t)P(Y≥t)=e−(λ+μ)t

So ZZ is an exponential random variable with parameter λ+μλ+μ.


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