Let $x_1, \ldots, x_n$ be a realization of a random sample $X_1, \ldots , X_n$ from continuous distribution. Then why $L(\theta)=f_\theta(x_1)\ldots f_\theta(x_n)$ holds? $f_\theta(x_i)$ means $P(X_i<x_i)$ which may overlap with another probability for some $k<i$. Does this hold because $X_i$ are independent?
2026-03-26 06:17:34.1774505854
Likelihood for continuous sample
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