Maximum of two exponentials as a scaled sum ($\max (X,Y)\stackrel{d}{=}X+Y/2$)

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.