= Self-decomposable distribution
{title2=$X\overset d=cX+R_c,\quad R_c\perp X$}
{wiki}
A <probability distribution> is self-decomposable if, for every $0<c<1$, an independent remainder $R_c$ can be chosen so that the displayed equality in <distribution> holds. A <normal distribution> $\mathcal N(\mu,\sigma^2)$ admits a <normal distribution> remainder with <mean> $(1-c)\mu$ and <variance> $(1-c^2)\sigma^2$. For an <exponential distribution> of rate $\lambda$, choose $R_c=0$ with <probability> $c$ and otherwise an independent <exponential distribution> of rate $\lambda$. Its <moment-generating function> is $c+(1-c)\lambda/(\lambda-t)=(\lambda-ct)/(\lambda-t)$; multiplication by $M_X(ct)$ proves the claim using the <uniqueness theorem for moment-generating functions>.
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