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Math Help - Easy question about creating a probability distribution

  1. #1
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    Easy question about creating a probability distribution

    Hi,
    I have a question about creating a probability distribution.:

    Let's say we have two sets of events with five elements (real numbers) and two random variables X and Y. So, we have one set for each random variable.
    In this case, we assign given probabilities to events from these sets.

    I am given the following question "find the probability distribution of XY."

    Could you explain me what steps I should take in doing this? I tried to multiply each event from X with each event from the other set to find XY and tried to compute the probabilities for each of the 25 events by multiplying the ones from the set of X with the ones from set of Y. However, I was not successful, and I got a total probability over 1. Does this sound too awful? Did I make a mistake here?

    I will be really glad, if you could help me to understand what the question means formally and what I should be doing in this situation.
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  2. #2
    Moo
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    Hello,

    Consider the values taken by X and the values taken by Y.
    Then find the possible values for XY, by multiplying the possibilities.

    Then find the probability to get a given value for XY.

    for example, if X can have value a with probability p, and Y can have value b with probability q, then P(XY=ab)=P(X=a,Y=b)
    And if X and Y are independent, then P(X=a,Y=b)=P(X=a)P(Y=b)
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  3. #3
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    Let me try to say it another way: in order to find P(XY=c), you should first find for which values of c this will be non-zero. In other words, find all the products that you can obtain by multiplying the values taken by X and Y. If X ranges from 1 to 5 and Y from 0 to 4, then the possible values for the product are 0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 20.

    Then for all possible value c, you have to decompose the event \{XY=c\} into the elementary events that describe all the outcomes for X and Y that would give a product equal to c, and sum their probabilities.

    For instance, if X and Y have positive integer values, P(XY=6)=P(X=1, Y=6) +  P(X=6,Y=1)+P(X=2,Y=3)+P(X=3,Y=2) (of course you can rule out a few cases if you know for instance that X can't be equal to 2).

    If you assume that X,Y are independent, then you have P(X=a, Y=b)=P(X=a)P(Y=b) for each term.
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