Probability distribution of a sum of random variables

Question

Suppose we have two independent random variables $X$ and $Y$. Their sum is $Z = X + Y$. How can I calculate the conditional distribution of $P(Z|X)$?

Answer 1

Recall that the conditional distribution of $Z$ conditioned by $X$ is defined as any family of probability measures $(\mu_x)_x$ such that, for every bounded function $u$, one has $E[u(Z)\mid X]=v(X)$, where the function $v$ is defined as $$ v(x)=\int u(z)\mathrm d\mu_x(z). $$ Here, one can take for each $\mu_x$ the distribution of $x+Y$. Thus, introducing the distribution $\nu$ of $Y$, for every Borel subset $B$ and every $x$, one sets $$ \mu_x(B)=P[x+Y\in B]=\nu(B-x). $$