Computing the density function of max of two items.

Question

Let $X$ be exponentially distributed with mean $3$. Then how do we compute the density function of $max\{X,2\}$?

Answer 1

Let $Z = \max(X,2)$. We calculate $P(Z > c)$. If $c < 2$, this is clearly $1$. If $c \geq 2$, then $Z=X$ so this is just $P(X > c)$.

This specifies the CDF of $Z$ (by $P(Z \leq c) = 1- P(Z>c)$). Now, differentiate it, and put a point mass at the jump at $2$ based on the height of the jump in the CDF.