Question:
If a population has exponential distribution with mean 1, then its median is:
If a population has exponential distribution with mean 1, then its median is:
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The median of an exponential distribution with mean 1 is given by \( \log_e 2 \), derived from setting the CDF equal to 0.5.
Updated On: Dec 2, 2025
- \( e \)
- 1
- \( \log_e 2 \)
- \( \log_e 3 \)
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The Correct Option is C
Solution and Explanation
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).
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