1. ## MLE

Hi guys, I need advice regarding this question:

The height of students at university is a random variable from the N($\displaystyle \mu$,25) distribution.
A sample of 100 students was taken, and it was found that k of them have a height below 175 cm.
a. based on this information, find a maximum likelihood estimator for $\displaystyle \mu$ .
b. calculate the MLE when k=33.

I have no idea how to approach this, anyone got an idea ?

thanks

2. MLE of what?
You need a parameter.
I would guess you want the MLE of $\displaystyle \mu$
if we know $\displaystyle \sigma^2$ in the normal setting.

3. sorry ! the Math didn't work at first, I fixed it now. Yes, I need an MLE for $\displaystyle \mu$ !

4. The MLE is $\displaystyle \mu$ is $\displaystyle \overline X$

5. Matheagle: The data appears to be censored, so you can't calculate $\displaystyle \bar{X}$.

My intuition says that you want to choose $\displaystyle \hat{\mu}$ so that the percentage of the sample under 175 is in line with what you would expect to see if you are sampling from $\displaystyle N(\hat{\mu}, 25)$. For example, if you 97.5% of your sample is under 175, you would set

$\displaystyle \frac{175 - \hat{\mu}}{5} = 1.96$.

That's just my intuition though, I haven't done any math.

Generalizing a bit, it looks like

$\displaystyle \hat{\mu} = 175 - (5)\Phi^{-1} (\alpha)$

where $\displaystyle \alpha = k/100$ and $\displaystyle \Phi$ is the CDF of the normal with mean 0 and variance 1. If k = 33, this turns out to make $\displaystyle \hat{\mu} = 177.2$. The intuition behind this is if we had an large number of observations and we found that (for example) the proportion less than 175 was exactly .5, we would have reason to believe that $\displaystyle F(175) = \Phi\left( \frac{175 - \mu}{5} \right) \approx .5$, where F is the CDF of our $\displaystyle N(\mu, 25)$.

Mathematically the justification for this is that you can think of $\displaystyle k \sim Bin\left(100, \Phi\left(\frac{175 - \mu}{5}\right)\right)$. We know how to find the MLE of a success probability, so its just a matter of solving for $\displaystyle \mu$.