Question

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

• A. \( P(a \leq X \leq b) = F(b) - F(a) \)
• B. \( F(x) \leq F(y) \text{ if } x \lt y \)
• C. \( \lim_{x \to -\infty} F(x) = 0 \text{ and } \lim_{x \to \infty} F(x) = 1 \)
• D. \( F(1) = \frac{1}{2} \)

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.

Answer 1

The Correct Option is D

Concept: The Cumulative Distribution Function (CDF) of a real-valued random variable \( X \), denoted by \( F(x) \), is defined mathematically as: \[ F(x) = P(X \leq x) \] Every valid probability cumulative distribution function must satisfy these fundamental mathematical properties:

• Monotonicity: \( F(x) \) is a non-decreasing function. If \( x \lt y \), then \( F(x) \leq F(y) \).

• Asymptotic Bounds: The limiting values at the extremes of the real line are bounded by absolute certainty scales: \[ \lim_{x \to -\infty} F(x) = 0 \quad \text{and} \quad \lim_{x \to \infty} F(x) = 1 \]

• Interval Probability: For any two real constants interval boundaries \( a \leq b \), the probability that \( X \) falls within the interval is given by: \[ P(a \lt X \leq b) = F(b) - F(a) \] (For continuous random variables, strict inequalities yield identical calculation values).

Step 1: Analyzing option (A).
The statement states \( P(a \leq X \leq b) = F(b) - F(a) \). This formula matches the fundamental definition of calculating interval probabilities from a cumulative distribution function. Therefore, this is a core property of a CDF.

Step 2: Analyzing option (B).
The statement states \( F(x) \leq F(y) \text{ if } x \lt y \). This is the definition of a non-decreasing (monotonic) function. Since cumulative probabilities can only accumulate or stay constant as you move from left to right along the real number line, this property is mathematically valid for all distribution functions.

Step 3: Analyzing option (C).
The statement specifies the limiting bounds: \[ \lim_{x \to -\infty} F(x) = 0 \quad \text{and} \quad \lim_{x \to \infty} F(x) = 1 \] As \( x \rightarrow -\infty \), the event \( X \leq x \) becomes an impossible event, so its probability approaches 0. As \( x \rightarrow \infty \), the event \( X \leq x \) becomes a certain event, so its probability approaches 1. This is a fundamental defining property of any valid cumulative distribution function.

Step 4: Analyzing option (D).
The statement asserts that \( F(1) = \frac{1}{2} \). This specifies that the cumulative probability exactly at \( x = 1 \) must always be equal to \( \frac{1}{2} \) for every random variable. However, the value of \( F(1) \) depends entirely on the specific probability distribution of the random variable \( X \). For example:

• For a uniform distribution on the interval \( [0, 10] \), \( F(1) = \frac{1}{10} \neq \frac{1}{2} \).

• For a standard normal distribution, the median is at 0, meaning \( F(0) = \frac{1}{2} \), so \( F(1) \approx 0.8413 \).
Since \( F(1) = \frac{1}{2} \) is only true for specific distributions and is not a general property of all distribution functions, it is not a universal property. Thus, option (D) is chosen as the incorrect property.