GATE 2026 DA Q.54: solution and answer

GATE 2026 · DAQ.54 · 2 marks · MSQ

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.

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Answer and solution

ANSWER

The correct statements are (B) and (C).

Official answer: B, CSource: Official answer key, GATE 2026 DA (IIT Guwahati)

Recall that the CDF is $F (x) = P (X \leq x)$ . It is non-decreasing, right-continuous, with $F (- \infty) = 0$ and $F (+ \infty) = 1$ .
(A) For any $x$ smaller than every value that $X$ can take, ${X \leq x}$ is empty, so $F (x) = 0$ . Zero is not a positive number, so $F$ is not always positive. (A) is false.
(B) If $a < b$ , then ${X \leq a} \subseteq {X \leq b}$ , so $F (a) = P (X \leq a) \leq P (X \leq b) = F (b)$ . Hence $F$ is non-decreasing. (B) is true.
(C) A discrete random variable takes some value $x_{0}$ with $P (X = x_{0}) > 0$ . At that point $F (x_{0}) - F (x_{0}^{-}) = P (X \leq x_{0}) - P (X < x_{0}) = P (X = x_{0}) > 0$ , so $F$ has a jump discontinuity of size $P (X = x_{0})$ . (C) is true.
(D) Because $F (x_{0}) = P (X \leq x_{0})$ includes the point mass at $x_{0}$ , we get $F (x_{0}) = F (x_{0}^{+})$ , so $F$ is right continuous. At a jump point, $F (x_{0}^{-}) = P (X < x_{0}) = F (x_{0}) - P (X = x_{0}) \neq = F (x_{0})$ , so $F$ is not left continuous. (D) is false.
The correct statements are (B) and (C).

KEY IDEA

Any CDF $F (x) = P (X \leq x)$ is non-decreasing and right continuous, and its jump at a point equals the point mass there. A discrete variable has such jumps, so $F$ has jump discontinuities and is not left continuous.

WATCH OUT FOR

Mixing up right and left continuity. Because $F$ uses $\leq$ , it is right continuous, not left continuous. Another slip is calling $F$ "positive" when it is only non-negative, since it equals $0$ to the left of the smallest value of $X$ .

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GATE 2026 DA Q.54: solution and answer

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