# Probability of the highest order statistic below the population median. [closed]

What is the probability that the highest order statistic of a random sample of size n from any continuous distribution is below the median ( population median ) of that distribution.

• What is the distribution? Uniform? Normal? Exponential? What? Commented Nov 26, 2014 at 0:15
• From any continuous distribution. (Question edited) Commented Nov 26, 2014 at 0:18
• no, we'd need to know what kind of distribution. The median and the probability of being below it, varies. Commented Nov 26, 2014 at 0:29
• This question is OK, nothing unclear about it. Should not be closed. Commented Jun 14, 2015 at 12:05
• @kjetilbhalvorsen The question is clear and totally lacking context.
– Did
Commented Jun 15, 2015 at 6:30

If the maximum is less than the median, then all points in the sample have to be less than the median. A sampled point will be less than the median with probability $1 \over 2$, so if $Y$ is the maximum and $m$ is the population median, then $$P[Y \le m] = {\left({1} \over {2} \right) }^n$$

• ... assuming the distribution is absolutely continuous, yes. Commented Jun 14, 2015 at 11:50

The probability that the highest order statistic in a sample of $n$ iid continuos random variables is below the median $m$ is:

\begin{align} \mathsf P(\max\{X_i\}_n<m) & = \mathsf F_X(m)^n \\[1ex] & = {\big(\tfrac 1 2\big)}^n & \text{by definition of the median} \end{align}

Here, $F_X$ is the cumulative probability function for the distribution, and by definition of what a median is.

• thank you for your time and response. Commented Nov 26, 2014 at 0:29
• Wrong! If the distribution is absolutely continuous, then $F_X(m)=1/2$, by definition of median. Commented Jun 14, 2015 at 11:51
• Why not delete?
– Did
Commented Jun 15, 2015 at 6:29