5.3.2 Deriving a Sampling Distribution

Probability rules can be used to obtain the distribution of a statistic provided that it is a “fairly simple” function of the $X_i$’s and either there are relatively few different $X$ values in the population or else the population distribution has a “nice” form.

Our next two examples illustrate such situations.

EXAMPLE 5.2.1

Example 5.21 should suggest first of all that the computation of $p_{\bar{X}}\left(\bar{x}\right)$ and $p_{S^2}(s^2)$ can be tedious. If the original distribution (5.2) had allowed for more than three possible values, then even for $n=2$ the computations would have been more involved. The example should also suggest, however, that there are some general relationships between $E\left(\bar{X}\right)$, $V\left(\bar{X}\right)$, $E\left(S^2\right)$, and the mean $\mu$ and variance ${\sigma }^2$ of the original distribution. These are stated in the next section. Now consider an example in which the random sample is drawn from a continuous distribution.

EXAMPLE 5.22