PDF question

[trickslapper]

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?

 

[tkhunny]

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?

 

[trickslapper]

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.