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By definition, it's stated that statistic said to be sufficient for θ if the conditional distribution of X1, X2, ..., Xn, given the statistic Y, does not depend on the parameter θ.

There is an examples provided below, I simply can't get why Y is not the sufficient statistic for this question.

Besides all the mathematical notations, would anyone give me an applied example of this? By the way, how this concept is related to the conjugate priors in Bayesian inference? Specifically, when it comes to estimating two unknown parameters. enter image description here

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