Let $X_1, X_2, \ldots , X_{999}$ be independent and identically distributed random variables on the interval $[-1/2, 1/2]$. Let $X_{500}$ be the empirical median; that is, $X_{500} = X_k$ for some $k$ such that for exactly 499 indices $j\neq k$ we have $X_j \leq X_k$ and exactly 499 indices $j \neq k$ we have $X_j \geq X_k$.
- Find an approximation for $P(X_{500}>0.01)$.
- What is the probability that $X_{500}=X_1?$
I'm not sure how to start this problem. Perhaps I should find $P(X_1+ \cdots+X_{499} \geq 499\cdot0.01)$?
We want the probability that at least $500$ of the $X_i$ are greater than $0.01$.
There are various ways to do this. The probability that an individual $X_i$ is greater than $0.01$ is $0.49$. Call an $X_i\gt 0.01$ a success. We want the probability that the number of successes in $999$ trials is $\ge 500$. A standard approach to this is the normal approximation to the binomial, with continuity correction.
For the probability that the empirical median is $X_1$, the empirical median is equally likely to be any of the observations.