# The joint pdf of sample maximum and sample mean for uniform distribution?

Assume $$\{X_i\}\stackrel{\mathrm{i.i.d.}}{\sim} \mathcal{Uniform}(0,1)$$ Find the joint p.d.f. of $$X_{(n)} \hat= \max \{X_1,X_2,\ldots,X_5\}\quad\text{ and }\quad \bar X\hat=\sum^n_{i=1}{X_i}$$

The motivation of this question: If a line of length 1 is randomly divided into five parts, computing the probability of there exist one part with length more than 1/4, i.e. find $$\Pr\{X_{(n)}\ge \frac{5\bar X}{4} \}$$ with the setting above.

• It is hard to see how $X_{(n)}\ge 5\bar{X}/4$ describes the event that there is one part with length more than $1/4.$ For instance, $X_{(n)}$ could be nearly zero and the other four values could all exceed $1/2,$ leaving an initial gap of $1/2.$ – whuber May 7 '19 at 22:08
• @whuber It's a typo, thank you. – C.X.Neo May 8 '19 at 7:11
• Please edit your post to display the expression you really want to evaluate, then! – whuber May 8 '19 at 14:11
• @whuber this question wants to find the joint p.d.f of $X_{(n)}$ and $\bar X$ – C.X.Neo May 9 '19 at 11:32

$$X_1, \dotsc,X_n$$ is iid from the uniform distribution on $$[0,1]$$. The maximum is $$X_{(n)}$$ which has a distribution with density $$f(u)=n u^{n-1}$$. Represent the joint density of the mean and the maximum as $$f(m, u)=f(m \mid u) f(u)$$ where $$f(m \mid u)$$ represents the conditional density of the mean given the maximum.
The order statistics $$X_{(1)},\dotsc, X_{(n-1)}$$, conditional on the maximum behaves as order statistics from an iid sample of size $$n-1$$ from the uniform distribution on $$[0,u]$$, so we can use that and the representation of the mean as $$\bar{X}_n = \frac{u}{n}+ \frac{n-1}{n}\bar{X}_{n-1}$$ when the maximum $$u$$ is given. Then the distribution of $$\bar{X}_{n-1}$$ is a scaled version of the Irwin-Hall distribution, see also Can we make the Irwin-Hall distribution more general?. I will leave it at that, you should be able to complete.