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Probability assignment

  1. Oct 2, 2011 #1
    1. The problem statement, all variables and given/known data

    Let Y be binary variable P(Y = 1) = P(Y = 0) = 0.5 and X a random variable uniform on [0,5] when Y = 0 and uniform [4, 9] when Y = 1. Draw mean of X and P(Y = 1|X = x) as functions of x. What is the minimum probability of rejection to predict Y from X without mistake.


    2. Relevant equations



    3. The attempt at a solution

    Since Y acts like some "switch", I considered two independant distributions of X given Y: p = 1/5 on [0,5] (for Y = 0) and the other one p = 1/5 on [4,9] (for Y = 1). So two means are: 2.5 (for Y = 0) and 7.5 for (for Y = 1).

    For P(Y = 1|X = x):
    if X in [0, 4]: P(Y = 1|X = x) = 0
    if X in [5, 9]: P(Y = 1|X = x) = 1
    if X in [4, 5]: P(Y = 1|X = x) = 0.5

    for this "What is the minimum probability of rejection to predict Y from X without mistake." I have no idea.
     
  2. jcsd
  3. Oct 2, 2011 #2

    vela

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    I think the problem is asking you for E[X] as a function of x, similar to what you did for P(Y=1|X=x). If x is in [0,4], what is E[X]? And so on.
     
  4. Oct 3, 2011 #3

    Ray Vickson

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    I don't see any way of predicting Y exactly from X in all cases. If we happen to observe a value of X between 4 and 5, Y is allowed to be 0 or 1, and there is no way to be sure which is correct.

    RGV
     
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