Let $p$ and $q$ be arbitrary probability mass functions of two discrete random variables. I need examples of functions $F(p,q)$ such that $r = F(p,q)$ and $r$ is a probability mass function for some random variable. Thank you.
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Let random variable $X$ take on value $a_i$ at the point $x_i$ with probability $p(x_i)$, and let random variable $Y$ take on value $b_j$ at the point $y_j$ with probability $q(y_j)$. Here for simplicity we are letting $i$ and $j$ range over the non-negative integers, but only minor modification needs to be made if one or both of the ranges is finite. Let $F(u,v)=uv$. Then $$\sum_{i,j=0}^\infty F(p(x_i)q(y_j))=\sum_{i,j=0}^\infty p(x_i)q(y_j)=\sum_{i=0^\infty}p(x_i)\left(\sum_{j=0}^\infty q(y_j)\right).$$ The inner sum is $1$, since $q$ is a probability mass function. But then the full sum is $1$, since $p$ is a probability mass function. So $F(p,q)$ has the critical property of being always non-negative and summing to $1$. If $X$ and $Y$ are independent, we can produce a random variable $Z$ closely connected to $X$ and $Y$ that has this distribution. Informally, then $F(p(x_i),q(y_j))$ gives us the probability that $X(x_i)=a_i$ and $Y(y_j)=b_j$. If $X$ and $Y$ are not independent, the connection with $X$ and $Y$ of the $Z$ we will produce is less close. The sample space of $Z$ is all ordered pairs $(x_i,y_j)$. In order to make sure that the values of $Z$ are real numbers, rather than the more natural ordered pairs of reals, let $\varphi: \mathbb{R}^2 \to \mathbb{R}$ be a fixed bijection. For any ordered pair in the sample space, $Z(x_i,y_j)$ has value $\varphi(a_i, b_j)$ with probability $p(x_i)q(x_j)$. There are more uninteresting examples that work. For example, we can let $F(u,v)=u$. I do not know of any general description of all the $F(u,v)$ that work. |
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