13

From a textbook:

Theorem 3.5. A function can serve as a probability density of a continuous random variable $X$ if its values, $f(x)$, satisfy the conditions^

  1. $f(x)\ge0$ for $-\infty <x<\infty$;

  2. $\int_{-\infty}^\infty f(x)\,dx=1$.

^The conditions are not "if and only if" as in Theorem 3.1 because $f(x)$ could be negative for some value of the random variable without affecting any of the probabilities. However, both conditions of Theorem 3.5 will be satisfied by nearly all the probability densities used in practice and studied in this text.

Could someone explain this further?

Thanks in advance

Michael Hardy
  • 1
Ahmed Ali
  • 1,116
  • 3
  • 13
  • 29
  • 5
    $f(x) = \begin{cases}1, & 0 < x \leq 1,\ -1, & x = 0,\0, &\text{otherwise,}\end{cases}$ can be used as the density function of a random variable uniformly distributed on $[0,1]$ without affecting probability calculations. – Dilip Sarwate Nov 25 '13 at 17:41
  • 2
    The conditions of Thm. 3.5 appear to aim at defining a probability density function on $(-\infty,\infty)$. In that case a negative value could be assigned on a set of measure zero, but it would not contribute anything to the definition of the probability distribution. – hardmath Nov 25 '13 at 17:47
  • 2
    So, short answer, no: probability density function should not be negative. – hardmath Nov 25 '13 at 17:48

1 Answers1

9

The text is trying to point out that changing a continuous probability distribution's density function at isolated points, even to negative values, does not change the probabilities defined by that density function.

More generally, once Lebesgue integration has been studied, we can speak of arbitrarily changing the values of the density function (integrand) on a set of measure zero, without however changing the integrals of that function.

As an example, consider the probability density function $f(x)$ which is zero for negative $x$ and $f(x) = e^{-x}$ for positive $x$. Then $f(0)$ can be any value we want, e.g. $f(0) = -1$, without changing the adequacy of $f(x)$ as a probability density function on $(-\infty,\infty)$.

hardmath
  • 37,015