# A Probability Problem Involving 6 Random Variables

1. May 11, 2008

### e(ho0n3

The problem statement, all variables and given/known data
Let $X_1, \ldots, X_6$ be a sequence of independent and identically distributed continuous random variables. Find

(a) $P\{X_6 > X_1 \, | \, X_1= \max(X_1, \ldots, X_5)\}$
(b) $P\{X_6 > X_2 \, | \, X_1 = \max(X_1, \ldots, X_5)\}$

The attempt at a solution
(a) is the probability that $X_6 = \max(X_1, \ldots, X_6)$ right? How would I determine this probability? In (b), the event $X_6 > X_2$ is independent of $X_1 = \max(X_1, \ldots, X_5)$ right? If it is, the probability is:

[tex]\int_{-\infty}^{\infty} \int_{x_1}^{\infty} f(x_6, x_1), \, dx_6 \, dx_1[/itex]

where $f(x_6, x_1) = f(x_6)f(x_1)$ since the random variables are independent. Right?

2. May 12, 2008

### e(ho0n3

I screwed up. $x_1$ should be $x_2$ in the above.