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Let $X_1 \cdots X_n$ be a random sample from a distribution $f(x) = e^{-(x-\theta)}, x>0, -\infty<\theta<\infty$. Let $Y_1 = \rm{min}\{X_1 \cdots X_n\}$.

Is $Y_1$ unbiased for $\theta$?

I'm stuck on finding the expected value of $Y_1$. I have the pdf as $f_{Y_1}(y) = n e^{-n(y-\theta)}$, but cannot find the integral $\int_\theta^\infty y n e^{-n(y-\theta)}$. I believe that integral is correct, but it appears to be undefined. Any idea where I may have gone wrong?

Let $X_1 \cdots X_n$ be a random sample from a distribution $f(x) = e^{-(x-\theta)}, x>\theta, -\infty<\theta<\infty$. Let $Y_1 = \rm{min}\{X_1 \cdots X_n\}$.

Is $Y_1$ unbiased for $\theta$?

I'm stuck on finding the expected value of $Y_1$. I have the pdf as $f_{Y_1}(y) = n e^{-n(y-\theta)}$, but cannot find the integral $\int_\theta^\infty y n e^{-n(y-\theta)}$. I believe that integral is correct, but it appears to be undefined. Any idea where I may have gone wrong?

Let $X_1 \cdots X_n$ be a random sample from a distribution $f(x) = e^{-(x-\theta)}, x>0, -\infty<\theta<\infty$. Let $Y_1 = \rm{min}\{X_1 \cdots X_n\}$.

Is $Y_1$ unbiased for $\theta$?

I'm stuck on finding the expected value of $Y_1$. I have the pdf as $f_{Y_1} = n e^{-n(y-\theta)}$, but cannot find the integral $\int_\theta^\infty y n e^{-n(y-\theta)}$. I believe that integral is correct, but it appears to be undefined. Any idea where I may have gone wrong?

Let $X_1 \cdots X_n$ be a random sample from a distribution $f(x) = e^{-(x-\theta)}, x>0, -\infty<\theta<\infty$. Let $Y_1 = \rm{min}\{X_1 \cdots X_n\}$.

Is $Y_1$ unbiased for $\theta$?

I'm stuck on finding the expected value of $Y_1$. I have the pdf as $f_{Y_1}(y) = n e^{-n(y-\theta)}$, but cannot find the integral $\int_\theta^\infty y n e^{-n(y-\theta)}$. I believe that integral is correct, but it appears to be undefined. Any idea where I may have gone wrong?