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    Probability distribution on next draw from random variable




    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:
    1. 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)

    2. (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 the true values 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, then 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 them that I need to solve for these functions.

    Thanks!

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    Re: Probability distribution on next draw from random variable


    This was stuck in the moderator queue and I just saw it. Do you still want help with this?
    I don't have emotions and sometimes that makes me very sad.

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