# Calculating probability based on mixed variables

Assume there are $K$ people and iid. parameters $a_1,\ldots,a_K$ associated to them with $a_i \sim U(0,1)$. Person $i$ observes his own fixed $a_i$ with some noise: $$X^{(1)}_i= a_i+ e^{(1)}_i,$$ where the $e_i$ is an error terms distributed normally with $N(0, v)$. He learns $X^{(1)}_i$.

After observing $X^{(1)}_i$, person $i$ would like to calculate the probability that next time the persons $1,\ldots,K$ get signals (denote the new signals by $X^{(2)}_i$), his signal will be greater than everyone else's signals, conditional on the value $X^{(1)}_i$.

Everyone's signal has an error term with the same distribution, and error terms are independent.

In summary, I would like to calculate the following:

$$\textrm{Prob}(X^{(2)}_i > X^{(2)}_{j}, \textrm{for all } j\neq i\mid X^{(1)}_i).$$

If someone can help me with how to approach this problem, I would really appreciate it.

• When you say that there is parameter $a$, do you actually mean that there are $K$ separate parameters $a_1,\ldots,a_K$? Are they independent? – Juho Kokkala Apr 16 '14 at 19:44
• For each person, there is a seperate $a_i$, yes. They are independent, but all come from standard uniform distribution $a_i \sim U[0,1]$ for all $i \in K$. – appletree Apr 16 '14 at 19:55
• Ok. I submitted an edit (which waits to be reviewed) to your question clarifying that point, and also using $\LaTeX$. – Juho Kokkala Apr 16 '14 at 20:18