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I have a random variable $X$ which is distributed uniformly over $[0,b]$. I want to know based on only one observation what is the confidence level of the interval $[X, 4X]$? I know this entails finding $P(X \leq b \leq 4X)$ but am entirely unsure how to approach the question. I have only had experience constructing confidence intervals for normally distributed variable and am very thrown off by the fact that $X$ is distributed uniformly.

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It's always the case that $X \leq b$, so you only need to find

\begin{align} P(b \leq 4X) &= P(1/4 \leq X / b) \\ &= 1 - P(X / b < 1/4) \\ &= 1 - 1/4 \\ &= 3/4 \end{align}

since $X / b \sim$ uniform$[0, 1]$.

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