The GATE Grind

GATE 2026 DA – Question 54

Probability and Statistics · Random variables, discrete and continuous distributions · 2 marks · Multiple select

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

Which of the following statements is/are correct?

  1. $F(x)$ is always a positive function.
  2. $F(x)$ is a non-decreasing function.
  3. $F(x)$ has jump discontinuity.
  4. $F(x)$ is a left continuous function.

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) $F(x)$ is a non-decreasing function.; (C) $F(x)$ has jump discontinuity.

Explanation

A cumulative distribution function never decreases (B is true). It is zero to the left of the smallest value of $X$, so it is not always positive (A is false). For a discrete variable it rises in jumps equal to the probabilities of the values, so it has jump discontinuities (C is true). A distribution function is right continuous, so at a jump it takes the upper value, which means it is not left continuous (D is false).