difference between convolution of two densities and mixture density?

Question

I am wondering about the difference of the convolution of two probability density functions and the mixture of those two. This is not the same right? But what is the difference and how can it be explained? I know that this gives different pictures, but I did not really understand convolution and therefore I cannot explain the difference of it to myself.

Answer 1

Say you have two independent random variables $X$ and $Y$, $X$ has density $f$ and $Y$ has density $g$. The convolution $f * g$ is the density of $X + Y$ while the mixture $\frac 1 2 f + \frac 1 2 g$ is the density of $W X + (1 - W) Y$ where $W$ is a Bernoulli $\mathcal B(\frac 1 2)$ independent of $X$ and $Y$.