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

The correct options is (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 ∈ (−∞, ∞) and (D): \(f(x) =   \begin{cases}     1-(In(x))^{-1},    & \quad 𝑥 ∈ (e, ∞)  \\     0,  & \quad   \text{ otherwis}   \end{cases}\)