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^
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$f(x)\ge0$ for $-\infty <x<\infty$;
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$\int_{-\infty}^\infty f(x)\,dx=1$.
^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
Best Answer
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 $(-\infty,\infty)$.