Solution Found!
Refer to Exercise 6.82. If Y1, Y2, . . . , Yn is a random
Chapter 6, Problem 83E(choose chapter or problem)
Refer to Exercise 6.82. If \(\mathrm{Y}_{1}, \mathrm{Y}_{2}, \ldots, \mathrm{Y}_{n}\) is a random sample from any continuous distributio with mean m, what is \(P(Y_{(n)}>m)\)?
Questions & Answers
QUESTION:
Refer to Exercise 6.82. If \(\mathrm{Y}_{1}, \mathrm{Y}_{2}, \ldots, \mathrm{Y}_{n}\) is a random sample from any continuous distributio with mean m, what is \(P(Y_{(n)}>m)\)?
ANSWER:Step 1 of 2
Given:
If Y is a continuous random variable and m is the median of the distribution, then m is such that \(P(Y \leq m)=P(Y \geq m)=1 / 2 .\)
If \(Y_{1}, Y_{2}, \ldots, Y_{n}\) are independent, exponentially distributed random variables with mean? and median m, Example 6.17 implies that \(Y_{(n)}=\max \left(Y_{1}, Y_{2}, \ldots, Y_{n}\right)\) does not have an exponential distribution. Use the general form of \(F_{Y_{(n)}}(y)\), and we will get \(P\left(Y_{(n)}>m\right)=1-(.5)^{n}\).