What is this distribution???

Question

Let $X_1, X_2, \ldots, X_n$ be a random sample from a population with $E(X_i) = \mu$ for all $i \in \{1,\ldots, n \}$.

Define

$ Y_i = \begin{cases} 1 & \mbox{ if } X_i < \mu \\ 0 & \mbox{ if } X_i > \mu \\ \end{cases} $

1) Determine the distribution of $Y = \sum_{i=1}^n Y_i$ (name and parameters)

Is there anyone who can give me a bit of guidance here. I think it's a binomial, but whatever $p = $ is unclear to me.

Answer 1

Hint:

The $Y_i$ are iid and Bernoulli distributed. The sum of $n$ such rv's is binomially distributed.

Answer 2

Exactly, this is binomial.

Hint: Each $Y_i$ is a Bernoulli random variable with two possible results (say $1=$ success if $X>μ$ and $0$-failure if $X <μ$. The sum of $n$ independent Bernoulli with the same success probability $$p=P(X>μ)$$ is binomially distributed with parameters $n$ and $p$, in symbols $$Y \sim \mathrm{Bin} (n, p)$$ where $p=P(X>μ)$.