Question

Order statistics help, please
\(Y_1~ Exp (1)\\Y_2~Exp(1.5\\Y_3~Exp(3)\)
Find \(P(X_{max} >3\)

Answer 1

What I don't get is:
\(P(Y_1\leq 3)=1-e^{-3}\\P(Y_2\leq 3)=1-e^{-3/1.5}\\P(Y_3\leq 3)=1-e^{-3/3}\)
Why?

Answer 2

@Zarkon

Answer 3

if \(X\sim \exp(\beta)\)

then \(f_{X}(x)=\dfrac{1}{\beta}e^{-x/\beta}\) for \(x\ge0\)

then \[\left.F_{X}(x)=P(X\le x)=\int\limits_{0}^{x}\dfrac{1}{\beta}e^{-t/\beta}dt=-e^{-t/\beta}\right|_{0}^{x}=-e^{-x/\beta}+e^{0/\beta}=1-e^{-x/\beta}\]

Answer 4

Let \[X=\max\{Y_1,Y_2,Y_3\}\]

\[P(X>3)=1-P(X\le 3)=1-P(\max\{Y_1,Y_2,Y_3\}\le 3)\]

if the maximum of the Y's is less than 3 then all the Y's are less than 3

\[=1-P(Y_1\le 3,Y_2\le 3,Y_3\le 3)\]

if the Y's are independent then we have

\[=1-P(Y_1\le 3)P(Y_2\le 3)P(Y_3\le 3)=\cdots\]

Answer 5

Thank you so much. I got it.