# Probability density function of $max(X,Y)$

Assume that we have a random variable $$W = \max({X,Y})$$ and that we would like to find the pdf of $$W$$. This is what I have done.

$$F_W(w)= \mathbb{P}[ W\leq w]=\mathbb{P}[ \max({X,Y})\leq w]=\mathbb{P}[ X\leq w]\mathbb{P}[Y\leq w]= F_X(w)F_y(w)$$ then the pdf is $$f_W(w) = \frac{dF_W(w)}{dw}=\frac{d (F_X(w)F_y(w))}{dw}= f_x(w)F_y(w)+ f_y(w)F_x(w)$$

Is my reasoning correct?

What if one want to find the distribution of $$W = \min({X,Y})$$?

• Assuming that $X$ and $Y$ are independent and both continuous random variables. Yes. Jan 22, 2015 at 3:03
• thanks, i want to find the min of the two variables, can you guide me through the process? Jan 22, 2015 at 3:12
• Same principle, only you want $f_Z(z) = \frac{\mathrm d\;\;}{\mathrm d\, z}\Big( 1-\Bbb P(\min(X,Y)>z) \Big)$ Jan 22, 2015 at 3:23
• Thanks for the answer, i dont know why i didnt think of it. Jan 22, 2015 at 3:32

$\color{green}{\checkmark}\quad$ Your reasoning for the density function of $W=\max(X,Y)$ is correct. Apply the same reasoning to find that of the minimum, with minor modification.
Vis: Let $Z=\min(X,Y)$ such that $X$ and $Y$ are independent and both continuous random variables, with density functions: $f_X$ and $f_Y$. (And cumulative distributions $F_X, F_Y$).
\begin{align} f_Z(z) & = \frac{\mathrm d\;}{\mathrm d z} \Bbb P(\min(X,Y)\leq z) \\[1ex] & = \frac{\mathrm d\;}{\mathrm d z} (1 - \Bbb P(\min(X,Y)\gt z)) \\[1ex] & = -\frac{\mathrm d\;}{\mathrm d z}\Big( \Bbb P(X>z)\,\Bbb P(Y>z) \Big) \\[1ex] & = -\frac{\mathrm d\;}{\mathrm d z} \Big(\big(1-F_X(z)\big)\big(1-F_Y(z)\big)\Big) \\[1ex] & = f_X(z)\Big(1-F_Y(z)\Big) + \Big(1-F_X(z)\Big)f_Y(z) \end{align}
• The event of the minimum being less than a number is the event of at least one variable being so.$$\{\min(X,Y)< n\}=\{X<n\}\cup\{Y<n\}$$However, the event of the minimum being greater than a number is the event of both variables being greater than that number. $$\{\min(X,Y)> n\}=\{X>n\}\cap\{Y>n\}$$ The probability for the intersection of two independent events equals the product of the probabilities for each event. So working with this is useful. Oct 18, 2020 at 6:26
• For maximum, this is the other way around. The event of the maximum being less than a number is the event of both variable being so. Thus this is the best to work with..$$\{\max(X,Y)< n\}=\{X<n\}\cap\{Y<n\}$$ Oct 18, 2020 at 6:30