Given $T$ iid random variables $X_1,X_2,\cdots,X_T$ (for $T\to+\infty$). How many random variables exceed the threshold $a$ (for some given $a$)? (Assuming that the random variables are positive.)
My work: let $Z_i$ be a binary variable that is $1$ if $X_i\geq a$. Thus, $\mathbb{E}[Z_i]=\Pr[X_i\geq a]$ and $$\sum_i\Pr[X_i\geq a]=T\Pr[X_1\geq a]$$ is the answer.
So, unless $\Pr[X_1\geq a]=0$, there are infinitely many random variables in the sequence that exceed the threshold $a$. Is this intuitive? I mean $a$ could be anything and still we have infinitely many.
If so, then, for $T$ iid exponentially distributed random variables, how could I choose $a$ to have $n$ random variables that are greater than $a$?
Best Answer
Because the random variables $X_i$ are i.i.d., for any fixed $a$ we have $\mathbb P(X_1\geqslant a)>0$, and hence $$ \sum_{i=1}^\infty \mathbb P(X_i\geqslant a) = \sum_{i=1}^\infty \mathbb P(X_1\geqslant a) = +\infty. $$ So by the second Borel-Cantelli lemma, $$ \mathbb P\left((\limsup_{n\to\infty} \{X_n\geqslant a\}\right)= 1. $$ In other words, $X_n\geqslant a$ for infinitely many $n$. Note that this is assuming that $\mathbb P(X_1\geqslant a)>0$. If this probability is zero, then this is not the case.