Random sum of random variables

Say you sum i.i.d. variables $X_i$ a total of $Y$ times. If you know the distribution of random variables $Y$ and $X_i$, what is the calculation you have to do to get the distribution of the sum?

If $S=\sum_{i=1}^{Y}X_{i}$, then the cumulant generating function of $S$ satisfies $$K_S(t)=K_Y(K_X(t)).$$ Any property of $S$ can be extracted from $K_S$.

How can one determine the chess configuration that maximizes the number of possible moves?

Dual of a holomorphic vector bundle

explicit cocycle representing Stiefel-Whitney class in Milnor and Stasheff

Prove that $\sum\limits_{i=0}^{k} p^{2i}$ ($p$ is prime) is never a perfect square

The coefficients of a series related to the Bernoulli polynomials

Mathematics Article Collection Books for Talented High School Students

covering space of $2$-genus surface

Generalizing the natural numbers - has this been studied before?

A Functional Differential Equation: $f^\prime(x) =\frac{f(2x)}{2f(x)}$

How many methods to this limit $\lim_{n\to\infty}\left(\frac{1}{2}+\frac{3}{2^2}+\cdots+\frac{2n-1}{2^n}\right)$

Deriving Fourier inversion formula from Fourier series

Coefficients in expansion of $(\sqrt[3]{2} - 1)^m$