5.3 Sampling Distribution

The sampling distribution of an estimator, ˆθ , follows a discrete distribution within the finite population setting. Thus the expected value of ˆθ defined by equation ?? can also be written as
   ˆ    ∑         ˆ
E (θ) =     k ⋅ P(θ = k )
         k
(5.2)

where

             ∑
P (ˆθ = k) =        P(s).
            ˆθ=k,s∈S

As an example using Table ?? where Ży is ˆθ , the expectation of Ży can also be determined using equation ?? as in Table ??.


Table 5.11: Sampling Distribution of Ży with an Unequal Probability Sampling WOR Design.



Ży P(Ży ) P(yŻ ) Ży



45.5 0.003 0.15
50.0 0.010 0.52
50.5 0.017 0.87
51.0 0.007 0.35
52.5 0.018 0.96
53.0 0.018 0.97
55.0 0.020 1.12
55.5 0.041 2.31
57.5 0.057 3.29
58.0 0.037 2.17
58.5 0.030 1.73
59.0 0.030 1.75
60.0 0.064 3.83
60.5 0.042 2.52
62.5 0.111 6.96
63.5 0.092 5.86
64.0 0.060 3.86
66.0 0.161 10.61
68.5 0.179 12.25



P(Ży ) Ży = 62.12 =E[Ży ]