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    fair dice rolled and X first value obtained which is <= 3 and Yis first value <=2 (so from 5, 5, 3, 6, 2 we have X=3 and Y=2)

    write down joint probability distribution of (X,Y) in table

    compute E(X) Cov(X,Y) E(XlY=1) and E(XlY=2)

    verify that E(E(XlY))=E(X)

    thanks
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    (Original post by number23)
    Suppose that a fair die is rolled and X is the first value obtained which is at most 3 and Y is first value at most 2.

    a) Write down joint probability distribution of (X,Y) in table.

    b) Compute E(X), cov(X,Y), E(XlY=1) and E(XlY=2).

    c) Verify that E(E(XlY))=E(X).
    I will only help you with part (a) as the rest of it is simply working through formulae and going through the notions.

    Split this into cases, suppose that X=Y in the first case.

    By construction then X=Y gives us two possibilities X=Y=1 or X=Y=2. Let n be the number of rolls we require to achieve this result and compute P(X=Y=1, n=p) where p is a positive integer.

    Recognising that each of the events which corresponds to these probabilities are mutually exclusive, we can take their sum and get P(X=Y=1) using geometric progressions.

    As we have a fair die then it's clear that P(X=Y=1)=P(X=Y=2).

    For the second case, recognise that P(X=x, Y=y) is non-zero if and only if X=3 when XY. We now apply a similar process as in the first case to yield the result.

    You should only need to compute 4 probabilities and of these 4, you only really need to do 2 of them as the die is fair. This cuts down on work significantly. All other probabilities should be 0.

    This should be enough to get through part (a). There may be more efficient ways to do this, but this is the most logical way I found to do it. I haven't done probability for awhile though, so there may be mistakes.

    I hope it does help though.

    Darren

    P.S.: Remember to check that your probabilities total to 1. I'm pretty sure they do.
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    (Original post by DPLSK)
    I will only help you with part (a) as the rest of it is simply working through formulae and going through the notions.

    Split this into cases, suppose that X=Y in the first case.

    By construction then X=Y gives us two possibilities X=Y=1 or X=Y=2. Let n be the number of rolls we require to achieve this result and compute P(X=Y=1, n=p) where p is a positive integer.

    Recognising that each of the events which corresponds to these probabilities are mutually exclusive, we can take their sum and get P(X=Y=1) using geometric progressions.

    As we have a fair die then it's clear that P(X=Y=1)=P(X=Y=2).

    For the second case, recognise that P(X=x, Y=y) is non-zero if and only if X=3 when XY. We now apply a similar process as in the first case to yield the result.

    You should only need to compute 4 probabilities and of these 4, you only really need to do 2 of them as the die is fair. This cuts down on work significantly. All other probabilities should be 0.

    This should be enough to get through part (a). There may be more efficient ways to do this, but this is the most logical way I found to do it. I haven't done probability for awhile though, so there may be mistakes.

    I hope it does help though.

    Darren

    P.S.: Remember to check that your probabilities total to 1. I'm pretty sure they do.
    thanks you so much! i was rlly having difficulty understanding this topic but will have a look through this
 
 
 
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