Find the median of the exponential random variable with parameter λ

Question

The median of a random variable X is a number µ that satisfies

Find the median of the exponential random variable with parameter λ.

Answer 1

For any positive number $x$, we have $$F_X(x)=\Pr(X\le x)=\int_0^x \lambda e^{-\lambda t}\,dt=1-e^{-\lambda x}.$$

So to find the median $m$, we want to solve the equation $$1-e^{-\lambda m}=\frac{1}{2}$$ for $m$, or equivalently to solve the equation $$e^{-\lambda m}=\frac{1}{2}.$$

Take the (natural) logarithm of both sides. We get $$-\lambda m=\ln(1/2)=-\ln(2).$$ Now solve for $m$. We get $$m=\frac{\ln 2}{\lambda}.$$

Remark: It is unfortunate that in the problem the symbol $\mu$ is used for the median, since $\mu$ is a standard name for the mean. A common symbol for the median is $m$.