Question

If a population has exponential distribution with mean 1, then its median is:

• A. \( e \)
• B. 1
• C. \( \log_e 2 \)
• D. \( \log_e 3 \)

Hint:

The median of an exponential distribution with mean 1 is given by \( \log_e 2 \), derived from setting the CDF equal to 0.5.

Answer 1

The Correct Option is C

The probability density function (PDF) of an exponential distribution with mean 1 is: \[ f(x) = e^{-x}, \quad x \geq 0 \] The cumulative distribution function (CDF) is: \[ F(x) = 1 - e^{-x} \] To find the median, we set the CDF equal to 0.5 (since the median is the value that divides the probability distribution in half): \[ F(x) = 0.5 \Rightarrow 1 - e^{-x} = 0.5 \] Solving for \( x \): \[ e^{-x} = 0.5 \quad \Rightarrow \quad -x = \ln(0.5) \quad \Rightarrow \quad x = \log_e 2 \] Thus, the median is \( \log_e 2 \), which corresponds to option (C).