Given that the lifetime of a certain type of lightbulb is distributed according to an exponential random variable with a parameter $\lambda$. You turn one light on, leave it in until it fails, and then immediately replace it with an exact identical lightbulb and leave that one until it fails. Find the probability density function for the total time the bulbs were on, and use it to calculate its expected value.
I have defined $X_i$ to be the random variable denoting the lifetime of the ith bulb.
I think that the probability density function of time lifetime should be the probability density function of $X_1+X_2$ but I am unsure of why mathematically that is.
Note: The lightbulbs are the exact same and have the same parameter $\lambda$.