Question

Which of the following functions qualify to be a cumulative density function of a random variable 𝑋 ?

• A. \(f(x) = \begin{cases}     1-e^{-x}       & \quad 𝑥 ∈ (0, ∞)  \\     0,  & \quad   \text{ otherwis}   \end{cases}\)
• B. 𝐹(𝑥) = (1 + 𝑒 ^−𝑥 ) ⁻¹ , 𝑥 ∈ (−∞, ∞)
• C. \(f(x) = \begin{cases}     1-x^{-1}in(x),       & \quad  𝑥 ∈ (e, ∞)  \\     0,  & \quad   \text{ otherwis}   \end{cases}\)
• D. \(f(x) = \begin{cases}     1-(In(x))^{-1},    & \quad 𝑥 ∈ (e, ∞)  \\     0,  & \quad   \text{ otherwis}   \end{cases}\)

Answer 1

The Correct Option is A, B, D

A cumulative distribution function (CDF) of a random variable \( X \) must satisfy the following properties:

1. The function \( F(x) \) is non-decreasing.
2. \(\lim_{x \to -\infty} F(x) = 0\) and \(\lim_{x \to \infty} F(x) = 1\).
3. \(F(x)\) is right-continuous. 

Let's evaluate the options given:

1. \(f(x) = \begin{cases} 1-e^{-x} & \quad x ∈ (0, ∞) \\ 0, & \quad \text{otherwise} \end{cases}\)
   • As \( x \to 0^{+} \), \( f(x) \to 0 \).
   • As \( x \to \infty \), \( f(x) \to 1 \).
   • This function is non-decreasing and bounded between 0 and 1.
   • Therefore, this is a valid CDF.

\( F(x) = \frac{1}{1 + e^{-x}}, \, x \in (-\infty, \infty) \)

• As \( x \to -\infty \), \( F(x) \to 0 \).
• As \( x \to \infty \), \( F(x) \to 1 \).
• This function is non-decreasing and bounded between 0 and 1.
• Therefore, this is a valid CDF.

1. \(f(x) = \begin{cases} 1-x^{-1}\text{in}(x), & \quad x ∈ (e, ∞) \\ 0, & \quad \text{otherwise} \end{cases}\)
   • As \( x \to e^+ \), this does not converge to 0.
   • This function may not be non-decreasing over its domain.
   • Therefore, this cannot be a valid CDF.
2. \(f(x) = \begin{cases} 1 - (\ln(x))^{-1}, & \quad x ∈ (e, ∞) \\ 0, & \quad \text{otherwise} \end{cases}\)
   • As \( x \to e^+ \), \( f(x) \to 0 \).
   • As \( x \to \infty \), \( f(x) \to 1 \).
   • This function is non-decreasing and bounded between 0 and 1 over its domain.
   • Therefore, this is a valid CDF.

Based on the evaluation, the functions that qualify to be cumulative density functions are:

• \(f(x) = \begin{cases} 1-e^{-x} & \quad x ∈ (0, ∞) \\ 0, & \quad \text{otherwise} \end{cases}\)
• \( F(x) = \frac{1}{1 + e^{-x}}, \, x \in (-\infty, \infty) \)
• \(f(x) = \begin{cases} 1 - (\ln(x))^{-1}, & \quad x ∈ (e, ∞) \\ 0, & \quad \text{otherwise} \end{cases}\)