[See] Suppose that the length of time Y it takes a worker to complete a certain task has the probability density function given by f(y)= \begincasese^-(y-θ),

Question: Suppose that the length of time $Y$ it takes a worker to complete a certain task has the probability

density function given by

\[f(y)= \begin{cases}e^{-(y-\theta)}, & y>\theta \\ 0, & \text { elsewhere }\end{cases}\]

where $\theta$ is a positive constant that represents the minimum time until task completion. Let $Y_{1}, Y_{2}, \ldots, Y_{n}$ denote a random sample of completion times from this distribution. Find

the density function for $Y_{(1)}=\min \left(Y_{1}, Y_{2}, \ldots, Y_{n}\right)$

b $E\left(Y_{(1)}\right)$

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[See] Suppose that the length of time Y it takes a worker to complete a certain task has the probability density function given by f(y)= \begincasese^-(y-θ),

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