Reserve Range

In thousands of dollars, let \(X\sim\operatorname{Uniform}(0,4)\) and \(Y\sim\operatorname{Uniform}(0,3)\) independently. Their joint density is \(\frac{1}{12}\) on the 4-by-3 rectangle. Let A denote \(X+Y\le4.5\). For each \(x\in[0,4]\), the accepted Y-length is \(w(x)=\min(3,4.5-x)\), which equals 3 on \([0,1.5]\) and \(4.5-x\) on \((1.5,4]\).

The total accepted area is
\[\int_{0}^{1.5}3\,dx+\int_{1.5}^{4}(4.5-x)\,dx=4.5+4.375=\frac{71}{8}\].
The accepted area with \(X>3\) is
\[\int_{3}^{4}(4.5-x)\,dx=1\].
Because the joint density is constant, it cancels in the conditional probability:
\[P(X>3\mid A)=\frac{1}{71/8}=\frac{8}{71}\].

An independent area check subtracts the excluded triangle and trapezoid from the full rectangle: the feasible area under the horizontal cap for \(x\le1.5\) is \(3\times1.5=4.5\), while the remaining trapezoid has parallel vertical sides of lengths 3 and 0.5 over width 2.5, area \(2.5(3+0.5)/2=4.375\). The numerator's trapezoid between \(x=3\) and \(x=4\) has heights 1.5 and 0.5, area \((1.5+0.5)/2=1\). The accepted widths shrink as x rises, so the conditional density of X is proportional to w(x), not a constant.