# Imagine $X_1$,…,$X_n$ are iid uniformly distributed and $X=\max(a_1X_1,..,a_nX_n)$, $Y=\max(b_1X_1,..,b_nX_n)$. What's the joint pdf of X and Y?

Imagine $$X_1$$,...,$$X_n$$ are i.i.d. uniformly distributed on the interval [0,1] and $$X=\max(a_1X_1,..,a_nX_n)$$ and $$Y=\max(b_1X_1,..,b_nX_n)$$ for some constants $$a_1,...,a_n$$, $$b_1,...,b_n$$.(All real, positive numbers)

What is the joint pdf (or cdf) of X and Y?

My idea: We have $$f_{X,Y}(x,y)=f_{X\mid Y}(x)f_Y(y)$$, Now the latter term can be calculated very easily because we have $$F_Y(x)=\prod_{i=1}^n F_{b_iX_i}(x)$$. So now we need to calculate the conditional probability;

However, I do not know how to continue here;

Any idea?

We have $$\Pr(Xwhere $$U(x)$$ is the cmf of uniform distribution on $$[0,1]$$. It is not generally easy to find the pmf from the above cmf by differentiation. It is all I got for now!
• Actually you must divide each $\min\{{x\over a_i},{y\over b_i}\}$ to the sections where $0<{x\over a_i}<1$ or $0<{y\over b_i}<1$. Also notice that the cdf is nonzero only if $0<\min\{{x\over a_i},{y\over b_i}\}<1$ for all $i$ – Mostafa Ayaz Nov 29 '18 at 10:06