In the problem I am trying to solve, I am asked to find the PDF of the second highest variable out of $U_1,U_2,U_2$ that are independent and follow the same uniform distribution in $[0,1]$. I originally thought that I can assume, without loss of generality, that $M_2=max\{U_1,U_2\}$ and $A_2=\{U_3>M\}$. Then, I've already found that for the maximum of two such variables and $W=max\{X_1,X_2\}$: $$f_W(w) = F_{X_1}(w)\cdot F_{X_2}(w)$$

So, is it correct to assume here that $F_M(m) = F_{U_1}(m)\cdot F_{U_2}(m) = m^2 \Rightarrow f_M(m) = F_M'(m) = 2\cdot m$ for $m\in[0,1]$ ?

Update: actually, I'm sure this is wrong because there are $m$'s for which $\mathbb{P} >1$, but I'm really lost...

  • 3
    $\begingroup$ It is the classical "order statistics" $X_{(2)}$. See here. Its pdf is $f(x)=kx(1-x)$ for convenient $k$. $\endgroup$
    – Jean Marie
    Commented Jan 10, 2022 at 22:50


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