Finding the MLE of parameter $\mu$

Question

Suppose $X_1,\ldots,X_n$ are a random sample of a continuous and strictly increasing distribution $F(x)$ with mean $\mu$. If

$$ Y_i = \begin {cases} 2 & \text{if}\ X_i>\mu \\ 1 & \text{if} \ X_i\leq\mu \end {cases} $$

Determine the MLE of parameter $\mu$

Answer 1

I would take a bet this is homework. Should give it the homework tag. As a hint, note that $$Pr(Y_i=1|\mu)=Pr(X_i\leq\mu|\mu)=F(\mu)=1-Pr(Y_i=2|\mu)$$

So the likelihood for the data is:

$$Pr(Y_1,\dots,Y_n)=\left[F(\mu)\right]^{n_1}\left[1-F(\mu)\right]^{n_2}$$

Where $n_1$ is the number of observed $Y_i$ which are equal to $1$ and $n_2$ is the number of observed $Y_i$ which are equal to $2$. Obviously $n_1+n_2=n$