= Maximum of two exponentials as a scaled sum
{title2=$\max(X,Y)\overset d=X+Y/2$}
For independent <exponential random variables> $X,Y$ with the same rate $\lambda>0$, the <convolution> density of $X+Y/2$ is $2\lambda(e^{-\lambda v}-e^{-2\lambda v})$ for $v>0$. The <cumulative distribution function> of $\max(X,Y)$ is $(1-e^{-\lambda v})^2$, whose derivative is the same density. This is equality of <probability distributions>, not an almost-sure equality of the two expressions formed from the same sample.
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