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If I have two different independent continuous variables, X and Y. Is it legitimate to multiply their pdf's to form a pdf for the ordered pairs (X,Y) as f(x,y).

I've plotted the surface and the volume under it is 1, and the key points (peaks and troughs seem to be in the correct position).

Is this a legitimate process? What issues do I need to be aware of?

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migrated from mathoverflow.net Feb 27 at 12:03

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yep, totally legit. –  Carlo Beenakker Feb 27 at 12:03
    
Yes, given you know/assume they're independent, it's correct. As for issues, there's not nearly enough information to guess at potential problems... it really depends on what you're doing, and why, and how. –  Glen_b Feb 27 at 21:58
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1 Answer 1

Multiplying the pdfs as you have done creates the joint pdf of the random variables $X$ and $Y$ or the pdf of the bivariate random variable $(X,Y)$ under the assumption that $X$ and $Y$ are independent random variables. Whether $X$ and $Y$ are indeed independent or not is something that you need to figure out from information that you have not provided to us. Knowledge of just the individual distributions of $X$ and $Y$ does not provide any information about independence.

As an example, suppose that $X$ and $Y$ are Bernoulli random variables with parameter $\frac{1}{2}$. Your multiplication procedure gives a joint distribution in which $(X,Y)$ takes on the $4$ values $(0,0), (0,1), (1,0), (1,1)$ with equal probability $\frac{1}{4}$. However, other joint distributions could also give you the same distribution for $X$ and $Y$, and in these other cases $X$ and $Y$ are not independent.

  • If $(X,Y)$ takes on values $(0,0)$ and $(1,1)$ with equal probability $\frac{1}{8}$ and values $(0,1)$ and $(1,0)$ with equal probability $\frac{3}{8}$, then $X$ and $Y$ are Bernoulli random variables with parameter $\frac{1}{2}$.

  • If $(X,Y)$ takes on values $(0,0)$ and $(1,1)$ with equal probability $\frac{1}{2}$, then $X$ and $Y$ are Bernoulli random variables with parameter $\frac{1}{2}$. Note that $X = Y$.

  • If $(X,Y)$ takes on values $(0,1)$ and $(1,0)$ with equal probability $\frac{1}{2}$, then $X$ and $Y$ are Bernoulli random variables with parameter $\frac{1}{2}$. Note that $X = 1-Y$.

In summary, the multiplication is valid if $X$ and $Y$ are independent random variables, but you need to provide some justification as to why you believe that $X$ and $Y$ are independent.

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