Matching a uniform and an exponential

EasyRandom Variables Statistics~4m

Let $X \sim \text{Uniform}(0, \lambda)$ and $Y \sim \text{Exponential}(\lambda)$ (rate $\lambda$ ), with $\lambda > 0$ . For what value of $\lambda$ do $X$ and $Y$ have the same variance? Give $\lambda$ to three decimal places.

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Recall $\operatorname{Var}(\text{Uniform}(0,\lambda)) = \lambda^2/12$ and, for the rate convention, $\operatorname{Var}(\text{Exponential}(\lambda)) = 1/\lambda^2$ .
Setting them equal gives $\lambda^4 = 12$ . Take the positive fourth root.

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1.861 (± 0.01)

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Matching a uniform and an exponential

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