Statistics question about PDFs.

Question

Let X be a U[0, 3] random variable (i.e., uniformly distributed on the interval [0, 3] ). Let Y be a U[0, 1] random variable independent of X. Let T = X + Y.

Find fT(t) = pdf of T (for all real numbers t ).

Answer 1

If you draw a picture of a rectangle of base 3 and height 1, the density is 1/3, and you can calculate the area of X+Y <= T by drawing a picture.

For 0 <= t <= 1, X+Y <= t is a triangle with base t and height t, so area 1/2t^2, and P(X+Y) <= t = 1/3(1/2t^2) = 1/6t^2

For 1 < t < 3, we have a trapezoid with bases t and t-1 and height 1, or area 1(t+(t-1))/2 = t - 1/2, and

P(X+Y) <= t = 1/3(t - 1/2) = 1/3T - 1/6

For 3 <= t <= 4, while the shape of X+Y <= t has 5 sides, the shape of X+Y >= t is a triangle, so we will make use of this. The base and height of the triangle are 4-t, so the area is 1/2(4-t)^2. Then, the area of X+Y <= t = 3 - 1/2(4-t)^2, and P(X + Y <= t) = 1/3(3 - 1/2(4-t)^2) = 1 - 1/6(4-t)^2

F(t) = 1/6t^2 0 <= t <= 1

1/3t - 1/6 1 <= t <= 3

1 - 1/6(4-t)^2 3 <= t <= 4

f(t) = F'(t)

f(t) = 1/3t 0 <=t <= 1

1/3 1 <= t <= 3

4/3 - 1/3t 3 <= t <= 4

Note how f(t) is an equilateral trapezoid with bases 4 and 2 and height 1/3, so area 1/3(4+2)/2 = 1