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Let $x$, $y$ and $z$ be independent random variables, uniformly distributed over the same interval. What is the cumulative distribution of $\max\{x+y,x+z\}$?

And more generally, let $x, y_1, y_2, ... y_n$, all be independent random variables, uniformly distributed over the same interval. What is the cumulative distribution of $\max\{x+y_1,x+y_2, ..., x+y_n\}$?

I wonder if the fact that $x$ being included in $w_i=x+y_i$ makes $w_i$ and $w_j$ interdependent.

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closed as off-topic by Did, kingW3, Shailesh, projectilemotion, Davide Giraudo Mar 9 '17 at 17:33

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  • $\begingroup$ Thoughts? Personal input? $\endgroup$ – Did Mar 9 '17 at 6:57
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Suppose $X$, $Y$ and $Z$ are iid Unif$(0,1)$,

then $T := \max(X+Y, X+Z) = X + \max(Y, Z)$. CDF of $W := \max(Y, Z)$ is $F_W(w) = \Pr(W \leq w) = \Pr(Y \leq w, Z \leq w) = \Pr(Y \leq w)\Pr(Z \leq w) =\begin{cases} 0 & \text{if } w\leq 0 \\ w^2 & \text{if } 0 < w< 1 \\ 1 & \text{if } w\geq 1 \end{cases} $ Therefore, density of $W$ is $f_W(w) = \begin{cases} 2 w & \text{if } 0 < w< 1 \\ 0 & \text{otherwise } \end{cases} $.

Since $X$, $Y$ and $Z$ are independent and $W = \max(Y, Z)$, so $X$ and $W$ are also independent. To find the distribution of $T = X+W$, we can use convolution:

$f_T(t) = \int\limits_{-\infty}^{\infty}f_X(x)f_W(t-x)dx = \begin{cases} \int\limits_{0}^{t}2(t-x)dx = t^2 & \text{if } 0 < t\leq 1 \\ \int\limits_{t-1}^{1}2(t-x)dx = 2t -t^2 & \text{if } 1 < t\leq 2 \\ 0 & \text{otherwise} \end{cases}$

CDF of $T$ is

$F_T(t) = \begin{cases} 0 & \text{ if } t\leq 0 \\ \frac{t^3}{3} & \text{if } 0 < t\leq 1 \\ t^2 - \frac{t^3}{3} - \frac{1}{3} & \text{if } 1 < t\leq 2 \\ 1 & \text{if } t > 2\end{cases}$

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  • $\begingroup$ Can you explain how you got the bounds for the integral in the convolution formula? $\endgroup$ – Parseval May 16 '18 at 13:10

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