Outline
In my own studies of probability theory, I came across the following explanation for deriving the PDF of the random variable $\max(X,Y)$. The methodology that they have outlined here seems unclear to me, as my intuition would be to first derive the CDF and then differentiate to obtain the PDF.
This method seems to evade the need to find the CDF, although the explanation is unclear.
The Explanation
Areas of Concern
The first area of concern is where the formula $\frac{b^2-a^2}{L^2}$ comes from (and what this quantity represents).
The second area of concern is how the PDF was obtained without any further working. The author implies that this is directly follows on from what is said, although I’m not sure that I understand how they have come to that conclusion.
I would be grateful for any guidance.
Thanks.
Best Answer
I am picturing that the square of side length $L$ is located in the plane with corners at $(0,0), (L,0), (L,L), (0,L)$.
At this point I would suggest letting $Z=\max(X,Y)$ (rather than using the author's notation where it looks like $x$ is used for a value of $\max(X,Y)$, which seems to me to be an over use of $x$ (and rather confusing)).
In order to have $a\le Z\le b$, (assuming $0\le a \le b\ \le L$), we must have both $X$ and $Y$ less than or equal to $b$ (inside a $b$ by $b$ square anchored at the origin), but not both less than or equal to $a$ (outside an $a$ by $a$ square anchored at the origin). This gives the area where $(X,Y)$ may be as $b^2-a^2$. Then relative to the entire area, we have $P(a\le Z \le b)=\frac{b^2-a^2}{L^2}$.
Then for the cdf, $P(Z\le b)=\frac{b^2}{L^2}$. Differentiating (as suggested in the footnote referenced in the comments) gives the pdf as $f_Z(b)=\frac{2b}{L^2}$ (valid for values of $b$ between $0$ and $L$; with the pdf being $0$ elsewhere).