The textbook I am using, using convolution in order to find the CDF of the $X+Y$ when $X\sim exp(\alpha)$ $Y\sim exp(\beta)$, and X and Y are are independent.
However, I have no background with convolutions at all (although my lecturer assumes I do, but that's another issue), so I am trying to figure this out without using convolutions, and I would really really appreciate if someone can point out what is wrong in my approach. of course I will also do my best to learn the convolution thing myself, but I am still very curious about my mistake here.
So:
Since both X and Y are non negative, I am only interested in the following region:
The numbers are arbitrary of course.
So my integration of the region is:
$$ F_Y(t)=P(Z\leq t)=P(X+Y\leq t)=\int_{0}^{t} \int_{0}^{t-x} \alpha e^{-\alpha x} \beta e^{-\beta y} dy dx $$
The result of the integration is this Wolfarm
While the convolution result is this Wolfarm
I really don't know what is so wrong about my work, can someone please help? Thanks!
In the convolution you have calculated the density of $Z = X + Y$ and in your result you essentially calculate the CDF of $Z = X + Y$. Differentiating your result with respect to $t$ leads indeed to the same result!
Since you are new in this field. Let me explain what happens in a discrete setting. For example, assume $X,Y\geq 0$ take on integer values and you want to know the distribution of $Z = X+Y$. Then what you generally do is write it as follows $$ P(Z=z) = P(X+Y = z) = P(X=z - Y) \\ = P((X=z,Y=0)\ \text{or}\ (X=z-1,Y=1)\ \text{or}\ \cdots\ \text{or}\ (X=0,Y=z)) \\ = \sum_{i=0}^{z}P(X=z-i,Y=i) = \sum_{i=0}^{z}P(X=z-i)P(Y=i). $$ On the left, we have the convolution of $X$ and $Y$. In a continuous setting, this would generalise to $$ f_Z(z) = \int_0^z f_X(z-y)f_Y(y)\,\mathrm{d} y. $$
I hope this makes it a bit more clear for you!