What are the expected values of the ordered arrival times

Question

Q:A store is expecting n deliveries between the hours of noon and 1 p.m. Suppose the arrival time
of each delivery truck is uniformly distributed on this one-hour interval and that the times are
independent of each other. What are the expected values of the ordered arrival times?

A:We consider the random uniform distribution of the delivery time taking place between 12 and 1. there are n deliveries in one hour.

The probability density function$\Rightarrow f(y_i)=1$
And the cumulative density function $\Rightarrow F(y_i)=\int_0^{y} 1\,d\tau=y_i$
$$F(y_i)=y_i,0\lt y_i\lt 1, and i=0,1,2,…,n $$
how can i find the expected values of the ordered arrival times?

Answer 1

Some clues:

Standard formula for order statistics: Suppose $X_, X_2, \dots, X_n$ is a random sample of size $n$ from a continuous distribution with density $f(x),$ where $f(x) > 0$ for $a < x < b.$ Then the density function of the $k$th order statistic $Y$ is $$\frac{n!}{(k-1)!(n-k)!}[F(y)]^{k-1}[1-F(y)]^{n-k}f(y),$$ for $a < y < b,$ and $0$ otherwise.

This implies that order statistics of $h$ standard uniform random variables will be beta random variables, with means evenly spaced at $1/(h+1), \dots, h/(h+1).$ For example if there are three of your arrivals in an hour, they will occur on average at 15, 30, and 45 minutes into the hour.

Simulation in R for the smallest of three:

set.seed(2020)
y1 = replicate(10^6, min(runif(3)))
mean(y1)> set.seed(2020)
[1] 0.2501943  # aprx E(min of 3) = 1/4

hist(y1, prob=T, br=50, col="skyblue2")
 curve(dbeta(x, 1,3),add=T, col="red", lwd=2)