Finding the Median and Mode of $X \sim \operatorname{Expo}(\lambda)$

Question

Question Let $X \sim \operatorname{Expo}(\lambda)$. Find the median and mode of $X \sim \operatorname{Expo}(\lambda)$.

My Solution:

Is my solution correct?

Answer 1

Your final answer is correct but your working is not.

• In your working to find the median, there is this line $$e^{-\lambda m} = -\frac12.$$ Exponential of any real number should be positive. You made a mistake involving the negative sign as you substitute the value in.

• In your working to find the mode, you claim that $\lambda=0$, but $\lambda$ is the parameter which should be positive. What you are interested is to find the value of $x$ that maximizes the probability density function. It need not satisfy $f'(x)=0$ as you claimed. It just have to be able to attain the maximum value which can occur at a boundary point.