Assuming that $H$, $N$ are random variables in which $H$ is distributed following exponential distribution with mean value $\Omega$, and $N$ is a Gaussian random variable with probability density function $$ f_N(x)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x^2}{2}\right). $$
Let $a$, $b$, and $c$ be constants. We define the following random variables
\begin{align} Y_1 &= |aH+bN|,\\ Y_2 &= |cN|. \end{align} Let $Y=\max\{Y_1, Y_2\}$. Find the CDF of $Y$.
Could you please give me a hint to find the CDF of $Y$. Thank you very much.
In general, if $Y_1$ and $Y_1$ are random variables and $Y=\max\{Y_1,Y_2\}$, then for any $x\in\mathbb R$, $$\{\omega: Y(\omega)\leqslant x\} = \{\omega :Y_1(\omega)\leqslant x\}\cap\{\omega: Y_2(\omega)\leqslant x\}. $$ So $$\mathbb P(Y\leqslant x) = \mathbb P\left(\{Y_1\leqslant x\}\cap\{Y_2\leqslant x\}\right). $$ If $Y_1$ and $Y_2$ are independent, then this reduces to $$\mathbb P(Y_1\leqslant x)\mathbb P(Y_2\leqslant x), $$ i.e. the product of the distribution functions of $Y_1$ and $Y_2$. If not, then it is a bit more complicated, and you would need to find the joint distribution of $(Y_1, Y_2)$.