Consider this question:
If $X$ is a random variable with $F_X$ (cumulative distribution function), then $X$ is absolutely continuous if:
a) $F_X(x)$ is differentiable for all $x$ in the real numbers.
b) $f_X(x)$ (Probability density function) is differentiable for all $x$ in the real numbers.
c) If the integral of $f_X(x)$ over all the real numbers is equal to $1$.
d) None of the above.
I know that the answer is d, but I don't know why or what even makes $X$ absolutely continuous.
Answer
In probability language, $F_X$ is absolutely continuous if there exists an associated probability density function $f_X$. This means there is the relationship $F_X(x)=\int_{-\infty}^x f_X(y) dy$. In this case we usually just say that $X$ is continuous, not absolutely continuous (though we may say something even more precise, like "the distribution of $X$ is absolutely continuous with respect to Lebesgue measure", if there is some possibility of confusion).
This does not require that $F_X$ be differentiable at every point; for a familiar example, you can look at $F_X(x)=\begin{cases} 0 & x<0 \\ 1-e^{-x} & x \geq 0 \end{cases}$. It certainly doesn't require $f_X$ to be differentiable at every point, indeed $f_X$ can even be nowhere continuous in principle.
Property (c) I don't entirely understand what is meant, seeing as absolute continuity is equivalent to $f_X$ merely existing.
No comments:
Post a Comment