I have the following problem. I want to find an expression for the following: $$\int_{0}^{\theta_1}\cdots\int_{0}^{\theta_{N-1}} \int_{0}^{\theta_N} \max\{X_1,...,X_N\}\hspace{1ex} dX_n\cdots dX_1$$
where $X_i$ $\overset{iid}{\sim} UNI[0,1]$, in addition we know that $0<\theta_1<...<\theta_N<1$. In the simple case where we have that $\theta_i = 1 $ $\forall i$ the expression reduces to $\frac{N}{N+1}$.
My approach so far was the following, which I already know doesn't work:
Define $Z = \max\{X_1,...,X_N\}$. obviously the support of $Z$ is $[0, \theta_N]$. But when trying to figure out the CDF of Z, there must be a mistake in my approach. So consider that $P(Z < u)$ $\overset{def}{=}$ $P(max\{X_1,...,X_N\} < u)$ $\overset{iid}{=}$ $P(X_1 < u)\cdot...\cdot P(X_N < u)$ $\overset{UNI}{=}$ $u^N$ . Therefore the density is given by $N \cdot u^{N-1}$. I know that this is wrong because the distribution of probability mass MUST depend on all the $\theta_i$s and not only on the amount of RV in the maximum function. I expect the density to be some kind of a trapezodial distribution with more than 4 parameters, starting at a high point, decreasing and the slope of it changing at each $\theta_i$.
So when continuing with the wrong approach, i get the following expression $$\int_0^{\theta_N}Z \cdot N\cdot Z^{N-1} dZ = \frac{N}{N+1} \theta_N^{N+1}$$ which is wrong.
Another thing which I was thinking about was that when deriving the CDF of Z I have to consider that in my particular expression the upper boundaries of the RV are not the same, even though the original RV are distributed iid. So $P(Z < u)$ $\overset{def}{=}$ $P(max\{X_1,...,X_N\} < u)$ $\overset{indep.}{=}$ $P(X_1 < u)\cdot...\cdot P(X_N < u)$ $= \frac{u}{\theta_1} \cdot \frac{u}{\theta_2} \cdot...\cdot \frac{u}{\theta_N} = \frac{u^N}{\theta_1 \cdot ... \cdot \theta_N} $. The densitiy is given by $\frac{N\cdot u^{N-1}}{\theta_1 \cdot ... \cdot \theta_N}$ such that the final expression is given by $$\int_0^{\theta_N}Z \cdot \frac{N\cdot Z^{N-1}}{\theta_1 \cdot ... \cdot \theta_N} dZ = \frac{N}{N+1} \frac{\theta_N^{N}}{\theta_1 \cdot ... \cdot \theta_{N-1}}$$
But this expression is also wrong when I compare it to matlab outputs for simple cases of two or three RV...
I look forward to helpful comments and thank you in advance.
In other terms, you want the integral of $\max(x_1,x_2,\ldots,x_n)$ over the (hyper-)rectangle $R$ having dimensions $\theta_1,\theta_2,\ldots,\theta_n$ with $\theta_1,\theta_2,\ldots,\theta_n$ being an increasing sequence in $[0,1]$. This sequence splits $[0,1]$ into $(n+1)$ sub-intervals $I_0=[0,\theta_1],I_1=[\theta_1,\theta_2],\ldots,I_n=[\theta_n,1]$ and if $u\in I_0$ the measure of $\{(x_1,x_2,\ldots,x_n)\in R:\max(x_1,x_2,\ldots,x_n)\leq u\}$ is exactly $u^n$, but if $u\in I_1$ the measure of such set is $\theta_1 u^{n-1}$, if $u\in I_2$ the measure of such set is $\theta_1 \theta_2 u^{n-2}$ and if $u\in I_k$ the measure of such set is $\theta_1\cdots\theta_k u^{n-k}$. It follows that the wanted integral equals
$$ \int_{0}^{\theta_0} un u^{n-1}\,du +\theta_1\int_{\theta_1}^{\theta_2} u(n-1) u^{n-2}\,du +\ldots + \theta_1\cdots\theta_{n-1}\int_{\theta_{n-1}}^{\theta_n} u\,du $$ or $$ \left[\frac{n}{n+1}u^{n+1}\right]_{0}^{\theta_1}+\theta_1\left[\frac{n-1}{n}u^n\right]_{\theta_1}^{\theta_2}+\ldots+\theta_1\cdots\theta_{n-1}\left[\frac{1}{2}u^2\right]_{\theta_{n-1}}^{\theta_n}. $$