Uniform and exponential sharing one parameter and mean

Question

For a constant $\lambda>0$ , let $X$ be uniform on $(0,\lambda)$ and $Y$ exponential with rate $\lambda$ . Find $\lambda^2$ for which $X$ and $Y$ have equal means.

Short answer

$2$ . The means are $\lambda/2$ and $1/\lambda$ ; setting them equal gives $\lambda^2=2$ .

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Lessons behind this question

Continuous Random Variables and Probability Density Functions
Cumulative Distribution Functions
Populations, Samples, and Sampling Distributions

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