Convolution or other methods for rv's transformation

Question

How can we solve using convolution or otherwise for the density function of $Z$ given that $Z=aX+bY$ where distribution of $X$ and $Y$ is known and $a$ and $b$ are constants. For example, of $X\sim Exponential (\lambda_1)$, $Y\sim Exponential (\lambda_2)$ and $Z=0.3X+0.7Y$. What if $X$ and $Y$ have different distributions with $X\sim U[0,2]$ and $Y\sim Exponential(\lambda)$.

I'm primarily struggle with the integration bit and it's intervals.

Answer 1

Convolution can be used only in particular cases. The main error in your question is that you did not assume independence between X and Y. If the rv's are not independent you cannot solve your problem unless you have enough information about the dependence structure.

The general method is to use the definition of CDF and integrate in the correct support.

Definition and a drawing are very useful.

$$F_Z(z)=\mathbb{P}[Z \leq z]$$

As an example let's set

$X\sim U(0;2)$

$Y\sim Exp(1)$

$X,Y$ independent

Calculate the law of $Z=0.3X+0.7Y$

Thus, by definition

$$F_Z(z)=\mathbb{P}[0.3X+0.7Y \leq z]=\mathbb{P}[Y\leq \frac{10}{7}z-\frac{3}{7}X]$$

...and with the useful help of a drawing

... you immediately see the domain to be integrated and thus

1. for $z<0$

$$F_Z(z)=0$$

2. For $0\leq z <\frac{3}{5}$

$$F_Z(z)=\int_{0}^{\frac{10}{3}z}\Bigg[\int_{0}^{\frac{10}{7}z-\frac{3}{7}x}\frac{e^{-y}}{2}dy\Bigg]dx$$

3. For $z \geq \frac{3}{5}$

$$F_Z(z)=\int_{0}^{2}\Bigg[\int_{0}^{\frac{10}{7}z-\frac{3}{7}x}\frac{e^{-y}}{2}dy\Bigg]dx$$

If you are interested in calculating $f_Z(Z)$ you have just to derivate $F_Z(z)$