Let $X,Y$ be i.i.d. random variables taking values in $\mathbb{Z_+}$. Suppose that either (a). $P(X=k\mid X+Y=n) = 1/(n+1)$ for all $0\leq k\leq n$,
or (b). $P(X=k\mid X+Y=n) = nC_k\cdot 2^{-n}$ for all $0\leq k\leq n$, provided $n$ is such that the condition has positive probabilities. Find the common distribution of $X$ and $Y.$
I am having some trouble understanding the solutions:
(a). define $\rho(n)=P(X=n)$ and $c_n=P(X+Y=n)/(n+1)$. The condition implies $\rho(k)\rho(l) = c_{k+l}$ for all $k,l\in \mathbb{Z_+}$. In particular if $\rho(m)>0$, then $\rho(k)>0$ for all $k\leq 2m$.
$\Rightarrow $I am really confused with the previous sentence: is $\rho(m)$ referring to $P(Y=m)$? I could not get anything close to $\rho(k)>0$ by using $\rho(k)\rho(l) = c_{k+l}$.
(continue with soln) Hence either $\rho(0)=1$, and thus $P\circ X^{-1}=\delta_0$, or $\rho(k)>0$ for all $k\in \mathbb{Z_+}$. In the second case $\rho(n)\rho(1)=\rho(n+1)\rho(0)$ for all $n\in \mathbb{Z_+}$.
Set $q=\rho(1)/\rho(0)$, by induction $\rho(n)=\rho(0)q^n$. As $\rho$ is a probability density, one has $\rho(0)=1-q$. Hence $X$ is geometrically distributed with parameter $p=\rho(0)$.
Any help on the latter part is also appreciated!
For a), let $X$ and $Y$ be independent geometrically distributed random variables, that is $\mathbb P(X=k)=(1-p)^{k-1}p, k=1,2,\ldots$. Then \begin{align} \mathbb P(X=k\mid X+Y=n) &= \frac{\mathbb P(X=k,X+Y=n)}{\mathbb P(X+Y=n)}\\ &=\frac{\mathbb P(X=k)\mathbb P(Y=n-k)}{\mathbb P(X+Y=n)}. \end{align} Now, \begin{align} \mathbb P(X+Y=n) &= \sum_{k=1}^n \mathbb P(X+Y=n\mid X=k)\mathbb P(X=k)\\ &=\sum_{k=1}^n \mathbb P(Y=n-k)\mathbb P(X=k)\\ &=\sum_{k=1}^n (1-p)^{n-k-1}p(1-p)^{k-1}p\\ &= p^2 \sum_{k=1}^n (1-p)^{n-2}\\ &= (n-1)(1-p)^{n-2}p^2, \end{align} so $$ \mathbb P(X=k\mid X+Y=n) = \frac{(1-p)^{k-1}p(1-p)^{n-k-1}p}{(n-1)(1-p)^{n-2}p^2} = \frac1{n-1},\quad k=1,\ldots,n-1 $$ Note that the probability is $\frac1{n-1}$ and not $\frac1{n+1}$ because $Y$ must take value at least $1$, so conditioned on $\{X+Y=n\}$, $X$ takes values in $\{1,2,\ldots,n-1\}$.