Problem 4

a) p_Uis a pdf, so the volume enclosed by it must equal 1, therfore k = 1/3.
b) The θ's are independent because it is a uniform distribution. The θ's must each cover all the data in the data set.

θ1 + 0.5 = 0.23 (xmax)
θ1 - 0.5 = -0.62 (xmin)
θ2 + 1.5 = 3.72 (ymax)
θ2 - 1.5 = 0.99 (ymin)

RowBox[{RowBox[{0.23, -, 0.5}], ≤, θ1, ≤, RowBox[{RowBox[{-, 0.62}], +, 0.5}]}] RowBox[{RowBox[{3.72, -, 1.5}], ≤, θ2, ≤, RowBox[{0.99, +, 1.5}]}]

RowBox[{RowBox[{-, 0.27}], ≤, θ1, ≤, RowBox[{-, 0.12}]}]

RowBox[{2.22, ≤, θ2, ≤, 2.49}]

Created by Mathematica  (October 8, 2003)