If I have a function
$$
F(x) = \left\{\begin{array}{}
0, & \text{if } x \leq -1\\
\frac{1}{2} – \frac{x^2}{2}, & \text{if } -1 \leq x \leq 0\\
\frac{1}{2} + \frac{x^2}{2}, & \text{if } 0 \leq x \leq 1\\
1, & \text{if } x \geq 1
\end{array}\right.
$$
how do I verify that it is a cumulative distribution function?
I know that to be a cumulative distribution function, $F$ must
- have $\lim_{x \to – \infty} = 0$,
- have $\lim_{x \to + \infty} = 1$,
- be non-decreasing,
- and be right-continuous.
Clearly $F$ satisfies the first two conditions by definition.
For $F$ to be non-decreasing, is it enough to show that $F'$ is always positive?
And I'm not sure even how to show that a function is right-continuous.
Would it be enough to sketch $F$, and then explain that it "looks like" a CDF?
Best Answer
Yes, showing that the derivative is non-negative everywhere is sufficient to show the function is non-decreasing. (When the derivative exists.)
To demonstrate right continuity of a piecewise function, show continuity within each piece and that the value at the left limit point of each piece is the limit of the piece from the right. Ie:
$$f(-1) = \lim\limits_{x\to -1^+} f(x) \\ f(0) = \lim\limits_{x\to 0^+} f(x) \\ f(1) = \lim\limits_{x\to 1^+} f(x)$$