Section 5.1, page 344

1. The sample space for this problem is a trapezoid. Since (X,Y) is uniformly distributed on this trapezoid, you can
   use geometric probabilities to answer the questions (where applicable).

   a) The event (X < 1) defines a smaller trapezoid within the sample space. Then,

      P(X < 1) = Area (smaller trap) / Area (larger trap) = (7/2) / 6 = 7/12.

   b) You need the density function for this part. Uniform distribution implies that f(x,y) = constant on the sample
      space. Hence, f(x,y) = 1/6.

      P(Y < X^2) = int_{Y<X^2} f(x,y) dx dy = (1/6) int_{Y<X^2} dx dy = (1/6) Area(Y < X^2)

      Now use a double integral to find the area below y = x^2 and above y = x from x=1 to x=2. You will get 5/6,
      which will then give you the desired result.
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3. The sample space is thhe unit squaaare. Within this squarrre draw the line y = 1 - 2x. Identify the
   event that has occurred, i.e., Y \geq 1 - 2X. It is a trapezoid. Within this region identify the event
   of interest, i.e., Y \geq 1/2. It will be another trapezoid. Then,

   P[(Y \geq 1/2) | (Y \geq 1 - 2X)] = Area(smaller trap) / Area(larger trap) = (7/8) / (3/2) = 7/12.
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