PDF question

Suppose that X[sub:2dhy8f4h]1[/sub:2dhy8f4h],..., X[sub:2dhy8f4h]m[/sub:2dhy8f4h] are independent exponential random variables with parameters ?[sub:2dhy8f4h]1[/sub:2dhy8f4h],...,?[sub:2dhy8f4h]m[/sub:2dhy8f4h], respectively. Determine and Identify the PDF of the random variable X=min{X[sub:2dhy8f4h]1[/sub:2dhy8f4h],...,X[sub:2dhy8f4h]m[/sub:2dhy8f4h]}

I kinda have an idea of what to do, but i'm not sure how the min comes into play. Any ideas how to attack this problem?

This is the fun question. If they had asked for Max(X1,X2, ...Xn) it would have been less fun.

Use your fundamental principles.

We have Pr(Xi>x) = exp(-x*li)

Pr(min(X1,X2, ...Xn)>x) =
Pr(X1 > x and X2 > x and ... and Xn > x)

Since they are independent.

Pr(X1 > x and X2 > x and ... and Xn > x) =
Pr(X1 > x)*Pr(X2 > x)*...*Pr(Xn > x)

Definition of individual PDFs

Pr(X1 > x)*Pr(X2 > x)*...*Pr(Xn > x) =
exp(-x*l1)*exp(-x*l2)*...*exp(-x*ln)

Log Rules

exp(-x*l1)*exp(-x*l2)*...*exp(-x*ln) =
exp(-x(l1+l2+...+ln))

One more thing to say. What is it?

umm i'm not sure is it that x>0 and the pdf is 0 otherwise?

I followed everything you did, but i thought i had to find the CDF and then take the derivative to come up with the PDF. I don't know my professor for this class has a hard time explaining these things since he can't really speak english that well.

thanks for the help though it is greatly appreciated.