The distribution density of a random variable $N=\max (X,Y)$

Question

Could you please help to solve the following problem:

Random variables $X,Y$ are independent and distributed exponentially with the same parameter $K$. Find the distribution density of a random variable $N=\max (X,Y)$.

Answer 1

In general, when $X$ and $Y$ are independent and identically distributed, $$P\{N \leq t\} = P\{X \leq t, Y \leq t\} = P\{X \leq t\} P\{Y \leq t\} = P\{X \leq t\}^2. $$ The first equality is because $\max\{x, y\}\leq t$ if and only if $x \leq t$ and $y \leq t$. The second equality uses the independence of $X, Y$, and the final equality uses that $X, Y$ have the same distribution.

Now since $X \sim \mathrm{Exp}(K)$, it follows that $P\{X \leq t\} = 1 - e^{-Kt}$, so that $P\{N \leq t\} = (1 - e^{-Kt})^2.$

Differentiating in $t$, the density of $N$, evaluated at $x$, is $ 2Ke^{-2Kx} \big(e^{Kx} - 1\big).$

Answer 2

In general $Z := \max(X, Y)$ is distributed as \begin{equation} F_Z(z) = \int_{-\infty}^z\,\int_{-\infty}^z\, f_{XY}(x, y)\, dx\, dy. \end{equation} Differentiation, using Leibniz's rule, gets you the density \begin{align} f_Z(z) &= F'_{Z}(z) \\ &= \int_{-\infty}^z\, f_{XY}(x, z)\, dx + \int_{-\infty}^z\, f_{XY}(z, y)\, dy. \end{align} In the case that $X$ and $Y$ are independent, $f_{XY}(x, y) = f_X(x)\, f_Y(y)$ and it follows $f_Z(z) = f_X(z)F_Y(z) + F_X(z)f_Y(z)$, where $F_X$ and $F_Y$ are the associated cumulative distribution functions.

So for $X$ and $Y$ independent and exponentially distributed, $f_X(x) = ae^{-ax}$ and $f_Y(y) = be^{-by}$ on $x > 0$, $y > 0$, you have $f_{Z}(z) = (1 - e^{-az})be^{-bz} + ae^{-az}(1 - e^{-bz})$, etc., which simplifies in your case $a = b$.