I have thought of a probabilistic model as follow:
$Y$ is Poisson($\lambda$)
$X|Y$ is Binomial(Y, 0.5)
Could you please tell me what is the name of this distribution of $X$ ? (i know it is a compound distribution but don't know the name). Moreover, does $X$ has a closed-formed distribution function ?
Thank you very much for your help!
First get the joint pmf
$$P(X,Y)=\frac{e^{-\lambda}\cdot \lambda^y}{y!}\binom{y}{x}\left(\frac{1}{2}\right)^y$$
Then sum w.r.t. y and after some manipulation you find that
$$P(X=x)=\frac{e^{-\lambda/2}\cdot(\lambda/2)^x}{x!}$$
That is $X\sim Po(\lambda/2)$
$$P(Y=y)=\frac{e^{-\lambda}\cdot \lambda^y}{y!}$$
the conditional distribution is
$$P(X|Y=y)=\binom{y}{x}\left(\frac{1}{2}\right)^y$$
the joint pmf is
$$P_{XY}(x,y)=P_Y(y)\cdot P_{X|Y}(x|y)$$