Question:easy

Which of the following is not the property of a distribution function \( F(x) \) of a random variable \( X \)?

Show Hint

When a question asks you to identify something that is ``NOT a property,'' look for options that specify an exact numerical value at a specific point (like \( F(1) = \frac{1}{2} \)). General properties must hold true for all possible distribution shapes, not just specific ones.
Updated On: Jul 4, 2026
  • \( P(a \leq X \leq b) = F(b) - F(a) \)
  • \( F(x) \leq F(y) \text{ if } x \lt y \)
  • \( \lim_{x \to -\infty} F(x) = 0 \text{ and } \lim_{x \to \infty} F(x) = 1 \)
  • \( F(1) = \frac{1}{2} \)
Show Solution

The Correct Option is D

Solution and Explanation

A cumulative distribution function \( F(x) = P(X \leq x) \) always has three guaranteed properties no matter what the underlying random variable is: it never decreases as \( x \) increases, it goes from \( 0 \) at \( -\infty \) to \( 1 \) at \( +\infty \), and the probability of landing in an interval is just the difference \( F(b) - F(a) \). Options (A), (B) and (C) are simply restating these three universal facts, so they always hold true for every distribution. The odd one out is option (D), which claims \( F(1) = \frac{1}{2} \) for every random variable. That cannot be a general rule since \( F(1) \) clearly depends on which distribution we are talking about. For example, for a random variable uniform on \( [0,10] \), \( F(1) = 0.1 \), and for an exponential random variable with rate \( 1 \), \( F(1) = 1 - e^{-1} \approx 0.632 \), neither of which is \( 0.5 \). So option (D) is not a property that holds in general, which makes it the answer to "which is not a property."
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