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Statistics for Economics Question on Probability theory

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

A

f(x)={1βˆ’eβˆ’xπ‘₯∈(0,∞)Β 0,Β otherwisf(x) = \begin{cases} 1-e^{-x} & \quad π‘₯ ∈ (0, ∞) \\\ 0, & \quad \text{ otherwis} \end{cases}

B

𝐹(π‘₯) = (1 + 𝑒 βˆ’π‘₯ ) βˆ’1 , π‘₯ ∈ (βˆ’βˆž, ∞)

C

f(x)={1βˆ’xβˆ’1in(x),π‘₯∈(e,∞)Β 0,Β otherwisf(x) = \begin{cases} 1-x^{-1}in(x), & \quad π‘₯ ∈ (e, ∞) \\\ 0, & \quad \text{ otherwis} \end{cases}

D

f(x)={1βˆ’(In(x))βˆ’1,π‘₯∈(e,∞)Β 0,Β otherwisf(x) = \begin{cases} 1-(In(x))^{-1}, & \quad π‘₯ ∈ (e, ∞) \\\ 0, & \quad \text{ otherwis} \end{cases}

Answer

f(x)={1βˆ’eβˆ’xπ‘₯∈(0,∞)Β 0,Β otherwisf(x) = \begin{cases} 1-e^{-x} & \quad π‘₯ ∈ (0, ∞) \\\ 0, & \quad \text{ otherwis} \end{cases}

Explanation

Solution

The correct options is (A): f(x)={1βˆ’eβˆ’xπ‘₯∈(0,∞)Β 0,Β otherwisf(x) = \begin{cases} 1-e^{-x} & \quad π‘₯ ∈ (0, ∞) \\\ 0, & \quad \text{ otherwis} \end{cases}, (B): 𝐹(π‘₯) = (1 + 𝑒 βˆ’π‘₯ ) βˆ’1 , π‘₯ ∈ (βˆ’βˆž, ∞) and (D): f(x)={1βˆ’(In(x))βˆ’1,π‘₯∈(e,∞)Β 0,Β otherwisf(x) = \begin{cases} 1-(In(x))^{-1}, & \quad π‘₯ ∈ (e, ∞) \\\ 0, & \quad \text{ otherwis} \end{cases}