# Relationships between random variables in a random sample

Suppose $$X_1, X_2, \ldots$$ are jointly continuous and independent, each distributed with marginal pdf $$f(x)$$, where each $$X_i$$ is nonnegative. Let $$T$$ be a random variable defined over the integers $$2, 3, \ldots$$ and let $$\Pr(T=i)$$ denote the probability that $$X_i > X_1$$ and $$X_j \leq X_1, \forall j.

a) Write out the explicit expression of $$\Pr(T=i)$$.

b) Prove that the expectation of $$T$$ is infinite.

My attempt

\begin{align} \Pr(T=i) &= \int_{x_1\geq x_2,\, \ldots, \,x_1\geq x_{i-1}, \,x_1

I'm not sure if this could be further simplified. About the expectation, I don't really have any idea how to proceed. A gentle hint would be appreciated. Many thanks in advance.

p.s. This is actually Exercise 5.2 from Casella & Berger.

It's much simpler than that. $$P(T=i)$$ is just the probability that $$X_i$$ is the first value in the sequence which is greater than $$X_1$$. $$P(T=2)$$ is trivially $$\frac{1}{2}$$, since $$X_2$$ is either greater or smaller than $$X_1$$ (the "continuous" bit tells us that there are no ties).
See also that the shape of $$f(x)$$ is not relevant; whatever the sequence, you can assign to each value of the sequence its relative rank. Consider a sequence of $$n$$. $$P(T=n)$$ is the probability that the biggest value is $$X_n$$, the last in the sequence, and the second biggest is $$X_1$$. So you simply have:
$$P(T=n) = \frac{1}{n}\frac{1}{n-1}$$