# Validity of Cumulative Distribution Functions & Probability Density/Mass Functions.

I have the following four CDF & PDF/PMF functions that I'm attempting to determine the validity of to ensure my understanding of the required conditions for these functions. I've tried to format these as best as possible and have provided my rationale and answer for each. I am unsure about the monotone non-decreasing requirement for the CDF. Is it true only for integers, i.e. $F(0) \le F(1)$ is required, or does it apply to the entire interval $[0,1]$? This greatly influences both the answers and my understanding. I've used the entire interval in my examples. I'm looking for someone to confirm my work, analysis and conclusions.

$$F(x) = \begin{cases} 0, &\text{if } x<0 \\ 4x^4 - 3x^2,&\text{if } 0\le x \le 1 \\ 1, &\text{if }x>1 \end{cases}$$

Rationale: $$\lim_{x \to\ -\infty} F(x) = 0 \ \ \checkmark \\ \lim_{x \to\ \infty} F(x) = 1 \ \ \checkmark \\ 4x^4 - 3x^2\text{ is monotone non-decreasing?} \ - FALSE$$

Answer: Since part 3 isn't monotone non-decreasing over the interval from $0$ to $1$, the CDF is invalid.

$$F(x) = \begin{cases} 0,&\text{if } x<-1 \\ x^2,&\text{if } -1\le x < 1 \\ 1,&\text{if } x \le 1 \end{cases}$$

Rationale: $$\lim_{x \to\ -\infty} F(x) = 0 \ \ \checkmark \\ \lim_{x \to\ \infty} F(x) = 1 \ \ \checkmark \\ x^2\text{ is monotone non-decreasing?} \ - FALSE$$

Answer: Since part 3 isn't monotone non-decreasing over the interval from $-1$ to $1$, the CDF is invalid.

$$f(x) = \begin{cases} \frac{1}{4}x^3, &\text{if } 0 \le x \le 2 \\ 0,& \text{otherwise} \\ \end{cases}$$

Rationale: $$f(x) \ge 0 \ \ \checkmark \\ \int_{-\infty}^{\infty} f(x)\, dx = \int_{0}^{2} \frac{1}{4}x^3\, dx = 1 \ \checkmark$$

Answer: Since both conditions hold true, the pdf/pmf is valid.

$$f(x) = \begin{cases} 6x-6x^2,&\text{if } 0 \le x \le 1 \\ 0,&\text{otherwise} \\ \end{cases}$$

Rationale: $$f(x) \ge 0 \ \ \checkmark \\ \int_{-\infty}^{\infty} f(x)\, dx = \int_{0}^{1} 6x-6x^2\, dx = 1 \ \checkmark$$

Answer: Since both conditions hold, the pdf/pmf is valid.

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Your answers look good. You might want to write $(1/4)x^3$ instead of $1/4x^3$ which could be misread as $1/(4x^3)$. Also, note that the first purported CDFs is negative for some $x$. – Dilip Sarwate Sep 25 '13 at 2:47