However, you should be aware that these "unconditional" moments do depend on

the initial distribution $N(\mu_0, \Sigma_0)$.

#### Moments of the Observations

#### Moments of the Observables

Using linearity of expectations again we have

However, there are some situations where these moments alone tell us all we

need to know.

These are situations in which the mean vector and covariance matrix are **sufficient statistics** for the population distribution.

These are situations in which the mean vector and covariance matrix are all of the **parameters** that pin down the population distribution.

(Sufficient statistics form a list of objects that characterize a population distribution)

One such situation is when the vector in question is Gaussian (i.e., normally

distributed).

The histogram and population distribution are close, as expected.

By looking at the figures and experimenting with parameters, you will gain a

feel for how the population distribution depends on the model primitives {ref}`listed above <lss_pgs>`, as intermediated by

the distribution's sufficient statistics.

feel for how the population distribution depends on the model primitives {ref}`listed above <lss_pgs>`, as intermediated by the distribution's parameters.

#### Ensemble Means