difference between convolution of two densities and mixture density? 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.
 A: 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 $Y$.
A: This is perhaps easiest to understand for discrete variables. For a convolution, the individual values of two random variables are paired and added. For a mixture distribution, there can be a single random variable, i.e., an $x_i$, which when enough values are accumulated in a histogram, follows the shape of two probability density functions that are scaled for AUC equals 1, when added together. 
For the continuous case, the convolution of two density functions is a density function. A convolution is an order independent operation sometimes symbolized $*$ or $\otimes $ that integrates two functions together as
$$\begin{align}
(f * g )(t) & \, \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^\infty f(\tau) g(t - \tau) \, d\tau \\
& = \int_{-\infty}^\infty f(t-\tau) g(\tau)\, d\tau
\end{align}\;,$$
that is, $(f*g)(t)=(g*f)(t)$, which means that one function can be thought of a smoothing of the other, and it does not matter which function is chosen as the smoothing function as the convolution process is commutative.  For example, the convolution of two exponential densities is a generic Bateman density, or an exponential density convolution (EDC), 
$$\mathrm{E}\mathrm{D}\mathrm{C}\left(b,\beta; t\right)=\mathrm{b}{e}^{-bt} \otimes \beta {e}^{-\beta t} = \left\{\begin{array}{l}\left.\begin{array}{l}\mathrm{b}\beta \frac{e^{-\beta t}-{e}^{-bt}}{\mathrm{b}-\beta },\ \mathrm{b}\ne \beta \\ {}\kern1.75em {\mathrm{b}}^2t\ {e}^{-\mathrm{b}t}\kern0.5em ,\ \mathrm{b}=\beta \end{array}\right\}t\ge 0\\ {}\left.\kern3.75em 0\kern4.1em \right\}t<0\end{array}\right..$$
and the convolution of two gamma densities, I call a gamma density convolution. A mixture model of two distributions is the arithmetic sum of those densities scaled so that the total area under the curve is 1 (which is a requirement for all density functions). For example for two exponential densities\begin{equation}
\begin{aligned}
\text{pdf}(t)&=p\text{ ED}(\lambda_1)+q\, \text{ED}(\lambda_2)\\&=p\lambda_1 e^{-\lambda_1 t}+q\lambda_2 e^{-\lambda_2 t}
\end{aligned}
\;,
\end{equation}
where $0<p<1$ and $q=1-p$ are the fractional (dimensionless) contributions of each ED to the pdf.
