# Distribution of variable sampled from bounded distribution?

First time asking question but have learned a lot reading through previous posts :)

Say I have a gym with 10000 members, so each day I will have a distribution of their time spent in the gym. This is obviously bounded from below by 0. If I take the person at the 80th percentile and track her for a year, what distribution will I get?

I'm not sure if it should just be some normal distribution centered at her time spent today. What about the 1st or 99th percentile? Since the time is bounded by 0, the 1st percentile likely doesn't form a normal distribution, and the 99th is probably some outlier and also doesn't.

• You are right that the Pr(X<0) > 0 if X ~ normal distribution in THEORY. Following this principle, could you list the variables following the normal distribution in PRACTICE? But we treat a lot of variables as normal distributed, for example, the weight of the people. Commented Jun 6, 2019 at 3:44

For samples of small or moderate size, each of the various order statistics of a distribution tends to have a distribution of a different shape. If you look at the (current version) of the Wikipedia article on order statistics you will see density functions of various order statistics of an exponential distribution; that's at the top right corner of the article.

Order statistics of a uniform distribution. Later on, the article says that the $$k$$th order statistic of a sample of size $$n$$ from $$\mathsf{Unif}(0,1)$$ has the distribution $$\mathsf{Beta}(k, n+1-k).$$ So for a a sample of size $$n = 4,$$ the figure below shows the density functions of the four order statistics.

As you can see from the Wikipedia link, there is a formula for the CDFs of distributions of the various order statistics from a given population. Sometimes, as in the case of a uniform population, results are easy to derive and compute.

Maximum of normal samples. Sometimes, it may be easier to do a simulation. The disstribution of the maximum of a sample of size $$n = 20$$ from a standard normal distribution is simulated below. As you can see, it is not normal.

set.seed(505)
x = replicate(10^5, max(rnorm(20)))
mean(x);  sd(x)
[1] 1.867599
[1] 0.5230468
summary(x)
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
0.139   1.501   1.825   1.868   2.188   4.944


More generally. However, there is a sort of "Central Limit Theorem" for quantiles of large samples from a wide variety of distributions. Conditions are that the quantile cannot be the maximum or the minimum, and that the density function of the population distribution must be positive at the quantile in question.

So, in particular, the median (50th percentile) of a sample of size 125 from $$\mathsf{Exp}(1)$$ is very nearly normal.

set.seed(1234)
x = replicate(10^5, median(rexp(125)))
mean(x);  sd(x)
[1] 0.6970988
[1] 0.08989155
summary(x)
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
0.3727  0.6342  0.6932  0.6971  0.7552  1.1954


• Thanks for the detailed answer! Commented Jun 13, 2019 at 1:58

The short answer is "we don't know, it depends on the distribution of time". BruceET in his answer (+1) gave some examples.

For your particular case, and going on what I know of gyms, the distribution of time spent in a gym is likely to be very odd. Not only is it bounded by 0, but there will be a lot of 0s. Then there will be a long right tail, but also with a somewhat mushy upper bound - just about no one spends more than 700 hours a year at the gym, except maybe professional bodybuilders and they tend to go to specialized gyms.

I think the way to approach this is to simulate various distributions for time spent at the gym and see what you get.