. Consider a stick of length one. Mark two points at random (that is,

two iid random numbers A, B in [0, 1]). This breaks the stick into three

segments. Let X, Y, Z be the lengths of the segments (in the order they

appear in the stick).

(a) Determine the probability that at least one of the marked points

is to the left of the midpoint.

(b) Determine P(X ≥ 1/2)

(c) Determine the density of Z.

(d) Determine P(X ≥ x, Y ≥ y).

(e) Determine the joint density of (X, Y ).

(f) Identify the distribution of (X, Y ).

(g) Determine the marginal density of X.

(h) Determine the conditional density of X given Y = y.

(i) Are X, Y mutually independent?

(j) Are X, Y, Z identically distributed?

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