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$$F=\left({\frac 1\theta}\right)^n\cdot e^{\sum_1^n x_i/\theta}$$ Taking logarithm, get $$-n\ln\theta-\sum_1^n x_i/\theta$$ Why do we get a minus sign in the second expression before the summation?

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Why do you have a minus sign there? Did you read this somewhere? If so, where? –  Jonas Meyer Dec 7 '12 at 3:13

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This looks like a joint density (or Likelihood) for $n$ iid Exponential random variables. If that is the case, then your first expression has a typo: there should be a minus sign before the sum in the exponent, and that's where the minus sign in the second expression comes from.

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