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Consider $X \sim \text{Unif} (\alpha, \beta)$. Find $P(X<\alpha + p(\beta - \alpha))$ Assume $p$ is a constant with $0<p<1$

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$X \sim Unif(\alpha, \beta)$

Find $\mathbb{P}(X<\alpha +\rho (\beta - \alpha))$ with $0<p<1$

Supposing that we have a continuous uniform distribution, we have the cumulative distribution function is $\mathbb{P}(X \le x) = \frac{x-\alpha}{\beta-\alpha}$ if $\alpha \leq x \le \beta$. This is just $\frac{\alpha +\rho (\beta - \alpha) - \alpha}{\beta - \alpha} = \rho$. Specifically, $\mathbb{P}(..) = 0$ for $\rho = 0$.

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