This statement is incorrect.

• The cumulative distribution function ($F(x)$) represents the probability $P(X \le x)$.
• Probabilities are always non-negative, meaning $F(x) \ge 0$.
• However, $F(x)$ is not always strictly positive. For values of $x$ less than the minimum possible value the discrete random variable $X$ can take, $F(x) = P(X \le x) = 0$.

This statement is correct.

• For any two values $x_1$ and $x_2$ such that $x_1 < x_2$, it must be true that $F(x_1) \le F(x_2)$.
• This means $P(X \le x_1) \le P(X \le x_2)$, which is always true because the event $\{X \le x_1\}$ is a subset of the event $\{X \le x_2\}$.
• This property holds for the CDF of any random variable, whether discrete or continuous.

This statement is correct specifically for a discrete random variable.

• For a discrete random variable, the probability mass is concentrated at specific points.
• The CDF, $F(x)$, increases in steps at these points.
• The size of the jump (discontinuity) at a specific value $x_0$ is equal to the probability that the random variable takes that exact value, i.e., $P(X=x_0)$.
• Mathematically, the jump at $x_0$ is given by $F(x_0) - \lim_{x \to x_0^-} F(x) = P(X=x_0)$. Since $P(X=x_0) > 0$ for the values $X$ can take, there is a jump discontinuity.

This statement is incorrect.

• The standard definition of a cumulative distribution function (for both discrete and continuous random variables) is that it is right continuous.
• Right continuity means that $\lim_{x \to a^+} F(x) = F(a)$ for any value $a$.
• For a discrete random variable, $F(x)$ has jump discontinuities, meaning it is generally not left continuous. At a point $a$ where $P(X=a) > 0$, we have $\lim_{x \to a^-} F(x) < F(a)$.