I'm working on a game theory model of incomplete information, where players observe certain attributes via noisy signals. I am looking to solve for two different probability functions, though I think the math should be very similar:
- Say there is some random variable $X \sim U(a - \epsilon, a + \epsilon)$. You observe a draw from this distribution, call it $S_1$, but you do not know $a$. Given $S_1$, what is the probability that another draw, $S_2$, from the same distribution, will be greater than or equal to an arbitrary number $b$? That is, what is:
$$P(S_2 \geq b \mid S_1)$$
- (Note these problems are separate, so, for example, $b$ here does not mean the same thing as $b$ in part 1). Say that $b$ and $c$ are two independent draws from $U(0,1)$, the standard uniform distribution. Now, $B \sim U(b - \epsilon, b + \epsilon)$ and $C \sim U(c - \epsilon, c + \epsilon)$. You observe one draw from $B$ and one draw from $C$, but you do not know $b$ or $c$. What is the probability that a new draw from $B$ will be greater than a new draw from $C$? That is, if $S_1^b$ is your observed draw from $B$ and $S_1^c$ is your observed draw from $C$, what is:
$$P(S_2^b \geq S_2^c \mid S_1^b, S_1^c)$$
Any help with this would be much appreciated! I've read a bit about convolutions and posterior predictive distributions, but I don't have the grasp on it that I need to solve for these functions.