Question

Three functions \(F_1(x)\), \(F_2(x)\) and \(F_3(x)\) are defined below:

(i) \(F_1(x) = 0\), for all \(x \in (-\infty, +\infty)\)
(ii) \(F_2(x) = 1\), for all \(x \in (-\infty, +\infty)\)
(iii) \(F_3(x) = 0\), for all \(x \le 0\) and \(F_3(x) = 1\), for all \(x > 0\)

Which of the above functions is a distribution function of a random variable?

• A. \(F_1(x)\) only
• B. \(F_2(x)\) only
• C. \(F_3(x)\) only
• D. None of these

Hint:

Check the three CDF axioms: limit 0 at \(-\infty\), limit 1 at \(+\infty\), and right-continuity at every point, especially at the jump point of \(F_3\).

Answer 1

The Correct Option is D

Step 1: Recall the defining properties of a cumulative distribution function (CDF) \(F\) of a random variable: \[\lim_{x \to -\infty} F(x) = 0, \qquad \lim_{x \to +\infty} F(x) = 1,\] \(F\) is non-decreasing, and \(F\) is right-continuous, meaning \(\lim_{x \to a^{+}} F(x) = F(a)\) at every point \(a\).

Step 2: Check \(F_1(x) = 0\) for all \(x\). Here \(\lim_{x \to +\infty} F_1(x) = 0 \ne 1\), so \(F_1\) fails the requirement that the limit at \(+\infty\) equals 1. \(F_1\) is not a valid CDF.

Step 3: Check \(F_2(x) = 1\) for all \(x\). Here \(\lim_{x \to -\infty} F_2(x) = 1 \ne 0\), so \(F_2\) fails the requirement that the limit at \(-\infty\) equals 0. \(F_2\) is not a valid CDF.

Step 4: Check \(F_3(x)\), defined as 0 for \(x \le 0\) and 1 for \(x > 0\). At the point \(a = 0\), the function value is \(F_3(0) = 0\) (since \(0 \le 0\)). But the right-hand limit is \[\lim_{x \to 0^{+}} F_3(x) = 1\] Since this right-hand limit (1) does not equal the function value at 0 (which is 0), \(F_3\) is not right-continuous at \(x = 0\). This violates the CDF requirement, so \(F_3\) is also not a valid distribution function (the correct degenerate-distribution CDF would instead be 0 for \(x < 0\) and 1 for \(x \ge 0\)).

Step 5: Since all three fail at least one CDF property, none of them is a valid distribution function.

The correct answer is None of these, option (D).