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Math Help - Joint density function of X(discrete) and Y(continuous)?

  1. #1
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    Exclamation Joint density function of X(discrete) and Y(continuous)?

    "Suppose X be a discrete random variable and Y be a continuous random varaible. Let f_X,Y (x,y) denote the joint density function of X and Y..."

    I'm not sure if I understand the meaning of joint density here. In the case above, what does the joint density mean? What are the properties of it? Does it still double integrate to 1 (like for the jointly continuous case)?

    From my probability class, I remember that a joint density function is only defined in the case when X and Y are BOTH continuous random variables. But when one random vaiable is continuous and the other discrete, how do we even define the joint density function?

    Hopefully someone can explain this.
    Thank you!
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  2. #2
    MHF Contributor matheagle's Avatar
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    You sum on the discrete one and integrate on the continuous
    I prefer to use the term density for only continuous RVs, but not everyone does that.
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    Quote Originally Posted by matheagle View Post
    You sum on the discrete one and integrate on the continuous
    I prefer to use the term density for only continuous RVs, but not everyone does that.
    Do you mean that
    ∫ ∑ f_X,Y (x,y) dy = 1 ?
    y x

    Is there an appropriate name for this kind of "density"?
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    Quote Originally Posted by kingwinner View Post
    Do you mean that
    ∫ ∑ f_X,Y (x,y) dy = 1 ?
    y x

    Is there an appropriate name for this kind of "density"?
    Mixed Distributions
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    But what defines a VALID density in this case?
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  6. #6
    MHF Contributor matheagle's Avatar
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    f(x,y)\ge 0 and that it sums/integrates to ONE
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    Quote Originally Posted by matheagle View Post
    f(x,y)\ge 0 and that it sums/integrates to ONE
    When you say it sums and integrates to one, do you mean
    ∫ ∑ f_X,Y (x,y) dy = 1
    y x
    where the sum is over the support of X and the integral is over the support of Y?

    Does it matter whether we do the sum first or the integral first? (i.e. can we interchange the ∫ and ∑ ?)
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  8. #8
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    Quote Originally Posted by kingwinner View Post
    When you say it sums and integrates to one, do you mean
    ∫ ∑ f_X,Y (x,y) dy = 1
    y x
    where the sum is over the support of X and the integral is over the support of Y? Mr F says: Yes.

    Does it matter whether we do the sum first or the integral first? (i.e. can we interchange the ∫ and ∑ ?) Mr F says: Most likely not. Review the appropriate theorems on when reversing the order of integral and summation is valid.
    ..
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