Is the sample sufficient? Assume that $ X_1 , ..., X_n $
is an i.i.d. sample from the population described by a density function $f_x (x,\theta) $ . 
Is the sample sufficient for $ \theta $ ?
I know it is a trivial question, but I am in doubt!
Thanks.
 A: Are you talking about sufficient statistics? If so, then you might be asking whether the identity function is a sufficient statistic. But note that the question is somehow ill-posed (read last paragraph).
The answer to this question is trivially yes: A statistic is sufficient if it provides at least as much information about a parameter as every other statistic you could compute from it. It basically means you throw nothing important away. The identity function throws nothing away, so it also doesn't throw anything important away.
But note that a statistic is a value that is obtained by applying some function to a sample. You asked whether a sample is sufficient, but a sample is not a value computed from a sample, so the question is somehow invalid. For example sample mean is a statistic and can be computed for a given sample. Sample mean is a sufficient statistic for the mean of a normal distribution. So given some sample $X_1,...,X_n$ from a normal distribution, it is enough to look at the sample mean when you are interested in the distribution mean. Of course the whole sample contains the information of the sample mean, so keeping the whole sample is also sufficient.
