The text I'm using on questions like these does not provide step by step instructions on how to solve these, it skipped many steps in the examples and due to such, I am rather confused as to what I'm doing.
Here is the question: Let $X$ be an exponential random variable with parameter $λ$ and $Y$ be an exponential random variable with parameter $2λ$ independent of $X$. Find the probability density function of $X + Y$.
Now, I know this goes into this equation: $\int_{-\infty}^{\infty}f_x(a-y)f_y(y)dy$
What I tried to do is $=\int_{-\infty}^{\infty}\lambda e^{-\lambda (a-y)}2\lambda e^{-\lambda y}dy$ but I quite honestly don't think this is the way to go. Can anyone give me a little insight as to how to actually compute $f_x(a-y)$ in particular?
Best Answer
$$\begin{align*} \Pr[X + Y \le t] &= \int_{x=0}^\infty \Pr[Y \le t - x \mid X = x] f_X(x) \, dx \\ &= \int_{x=0}^t (1 - e^{-2\lambda(t-x)}) \lambda e^{-\lambda x} \, dx \\ &= \lambda \int_{x=0}^t e^{-\lambda x} - e^{-2\lambda t} e^{\lambda x} \, dx \\ &= \left[ -e^{-\lambda x} - e^{-2\lambda t} e^{\lambda x} \right]_{x=0}^t \\ &= 1 + e^{-2\lambda t} - 2e^{-\lambda t}. \end{align*}$$
The probability density is then found by differentiation with respect to $t$.