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?

**Contents**hide

#### Best Answer

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.