Question

Let \(X\) be a discrete valued random variable with cumulative distribution function \(F(x)\).

Which of the following statements is/are correct?

• A. \(F(x)\) is always a positive function.
• B. \(F(x)\) is a non-decreasing function.
• C. \(F(x)\) has jump discontinuity.
• D. \(F(x)\) is a left continuous function.

Hint:

Check the CDF definition \(F(x)=P(X\leq x)\) directly against each claim: it is non-negative and non-decreasing always, and being right continuous with jumps is specific to discrete random variables.

Answer 1

The Correct Option is B, C

A cumulative distribution function (CDF) is defined as $F(x) = P(X \leq x)$. For a discrete random variable, this function is built entirely out of the probability masses at the individual points $X$ can take, which gives it a very specific step-function shape. Let's check each statement about $F(x)$ against this definition.

1. $F(x)$ is always a positive function: A CDF is a probability, so it can never be negative, but it is not always strictly positive either. For any $x$ below the smallest value $X$ can take, $F(x) = 0$ exactly, since $P(X \leq x) = 0$ there. So calling it "always positive" is too strong; the correct statement would be "always non-negative". This option is wrong.
2. $F(x)$ is a non-decreasing function: Take any $x_1 < x_2$. The event $X \leq x_1$ is entirely contained inside the event $X \leq x_2$, so its probability cannot exceed the probability of the larger event. That means $F(x_1) \leq F(x_2)$ always, for any distribution, discrete or continuous. This option is correct.
3. $F(x)$ has jump discontinuity: Because $X$ is discrete, all of its probability is concentrated at a countable set of points. Right at each such point, $F(x)$ suddenly increases by that point's probability mass, then stays flat until the next point. This staircase behavior is precisely a jump discontinuity at every value $X$ can take. This option is correct.
4. $F(x)$ is a left continuous function: The standard definition $F(x) = P(X \leq x)$ already includes the value $x$ itself, so nudging $x$ slightly to the right does not change what is included, making $F$ right continuous. Nudging $x$ slightly to the left, though, excludes the probability mass sitting exactly at $x$, so the left-hand limit falls short of $F(x)$ at any jump point. So $F(x)$ is right continuous, not left continuous. This option is wrong.

Only the non-decreasing property and the jump discontinuity property hold for a discrete random variable's CDF.

Let's summarize:

• A CDF is non-negative, not strictly positive everywhere.
• Monotonic non-decreasing behavior is a universal CDF property, for any type of random variable.
• Discreteness of $X$ is exactly what produces jump discontinuities in $F(x)$.
• A CDF is right continuous by convention, never left continuous at a jump.

So the correct statements are (B) and (C).