Question Bank • QB_EM_PS_005
MCQ • 2 Marks
Section 1: Engineering Mathematics
Probability and Statistics
Subtopic: Random variables
Question Statement:
The cumulative distribution function (CDF) $F(x) = P(X \le x)$ of any real random variable $X$ must satisfy which of the following properties?
Options:
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(A) $F(x)$ is non-decreasing, right-continuous, with $\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to \infty} F(x) = 1$
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(B) $F(x)$ is strictly increasing and left-continuous everywhere
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(C) $F(x)$ must be symmetric about $x = 0$
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(D) $F(x)$ is unbounded as $x \to \infty$
Step-by-Step Solution & Explanation
By axiomatic probability theory, every CDF $F(x)$ satisfies:
1. Non-decreasing: if $x_1 \le x_2$, then $F(x_1) \le F(x_2)$.
2. Right-continuous: $\lim_{h \to 0^+} F(x + h) = F(x)$.
3. Limits: $\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to \infty} F(x) = 1$.
Thus, option A is correct.
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