Minimum of random and constant scaled with random variable

Question

Suppose that $X$ and $Y$are independent random variables, while $c \geq 0$ is a constant. More specifically, $X$ is the $n$-fold convolution of $Y$. Is it then correct to say the following?

$\min(Y,c)+X = \min(Y+X, c+X)$

Obviously this would be correct for only constants. But does it hold for random variables?

Answer 1

This has nothing to do with random variables. For any three real numbers $a,b,c$ we have $\min (a,c)+b =\min (a+b,c+b)$ so the same holds for random variables. [You can just verify the equation for the cases $a \leq c$ and $a>c$].