Let f be a joint density function of a pair of continuous random variables X and Y. What properties does f possess?

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Rayna Aguilar · Accepted Answer

To be a proper probability density function, f must be normalized, i.e. the integral of f over the parameter space must be one.      ∫          −      ∞              ∞            ∫          −      ∞              ∞        f  (  x  ,  y  )  d  A  =  1The probability density function must be non-negative everywhere, because it is meaningless for an event to have probability less than one.  f  (  x  ,  y  )  ≥  0 for all(x,y)Result:Property 1 :  Since probabilities are non-negative values and are measured on a scale from 0 to 1, the int density function f is always non-negative i.e.  f  (  x  ,  y  )  ≥  0Property 2 : Since the sum of all probabilities is 1, hence  ∫      ∫                            R                          2                      f  (  x  ,  y  )  d  A  =      ∫          −      ∞              ∞            ∫          −      ∞              ∞        f  (  x  ,  y  )  d  x  d  y  =  1

Let f be a joint density function of a pair of continuous random variables X and Y. What properties does f possess?

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