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MTHE/STAT 353 – Winter 2016
Homework Assignment 2
Assignment 2 — due Friday, Jan. 22
1. Consider a square with edges labelled 1, 2, 3, 4. Each vertex of the square has a value
which is randomly chosen from the set {1, 2, 3, 4, 5, 6}, independently from vertex to
vertex. Let
1 if edge i connects two vertices with the same value
Xi =
0 otherwise.
Show that X1 , X2 , X3 , X4 are pairwise independent but are not mutually independent.
2. Suppose that the lengths, X1 , X2 and X3 , of three rods are independent uniform(0,1)
random variables. Determine the probability that it is possible to form a triangle from
the three rods. Hint: One cannot form a triangle if and only if one of the three rods is
longer than the sum of the lengths of the other two.
3. Suppose that X1 , . . . , Xn are independent and identically distributed continuous random variables, each with probability density function
3/(x + 1)4 if x ≥ 0
f (x) =
0
if x < 0
Find E[X12 X22 . . . Xn2 ].
4. Here’s a question that arises in coverage problems. Consider an equilateral triangle T
with side length d, and with one vertex labelled v. A point A is chosen at random
from a circle of radius r0 centred at v. Then a circle C of radius r is drawn centred
at A. Let X denote the area of the intersection of the triangle T and the circle
√ C.
Note that d, v, r0 and r are all fixed (not random). Assume that r0 + r < 3d/2.
Find E[X]. Hint: This question can be done in your head using symmetry (consider 6
triangles forming a hexagon). Note that for an equilateral triangle√with side length d,
the distance from a vertex to the midpoint of the opposite side is 3d/2.
5. Suppose that the random vector X = (X1 , X2 , X3 ) is uniformly distributed on the
sphere of radius r centred at the origin; that is, X has joint probability density function
3
if (x1 , x2 , x3 ) ∈ S
4πr3
fX (x1 , x2 , x3 ) =
0 otherwise,
where S = {(x1 , x2 , x3 ) ∈ R3 : x21 + x22 + x23 ≤ r2 } is the sphere of radius r centred at
(0, 0, 0). Let Y = X12 + X22 + X32 . Find the distribution function of Y , the probability
density function of Y , and E[Y ].
6. Let X1 , . . . , Xn be jointly continuous random variables with joint pdf f . Show that
X1 , . . . , Xn are mutually independent if and only if
E[g1 (X1 ) . . . gn (Xn )] = E[g1 (X1 )] . . . E[gn (Xn )]
for all functions g1 , . . . , gn for which the above expectations exist.
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