Download Homework 11

Math 151- Probability
Spring 2015 Jo Hardin
Homework 11 - Solutions
Assignment
[11] DeGroot, section 4.4
Suppose X is a random variable for which the m.g.f. is:
ψ(t) =
1 2t 2 4t 2 8t
e + e + e ,
5
5
5
−∞ < t < ∞.
Find the probability distribution of X. Hint: it is a simple discrete distribution.
[12] DeGroot, section 4.4
Suppose that X is a random variable for which the m.g.f. is:
ψ(t) =
1
(4 + et + e−t ),
6
−∞ < t < ∞.
Find the probability distribution of X.
[13] DeGroot, section 4.4 Let X have the Cauchy distribution, so the density of X is:
f (x) =
1
,
π(1 + x2 )
−∞ < x < ∞.
Prove that the m.g.f. of X is finite only for t = 0.
[6] DeGroot, section 4.5 Suppose that a random variable X has a continuous distribution for which the p.d.f.
f is:
(
2x for 0 < x < 1
f (x) =
0
otherwise .
Determine the value of d that minimizes
1. E[(X − d)2 ] (the mean squared distance),
2. E[|X − d|] (the mean absolute distance).
[9] DeGroot, section 4.6
Suppose that X and Y are two random variables which may be dependent and V ar(X) = V ar(Y ). Assuming
that 0 < V ar(X + Y ) < ∞ and 0 < V ar(X − Y ) < ∞, show that the random variables X + Y and X − Y
are uncorrelated.
[12] DeGroot, section 4.6
Suppose that X and Y have a continuous joint distribution for which the p.d.f. is
(
1
(x + y) 0 ≤ x ≤ 1 and 0 ≤ y ≤ 2
f (x, y) = 3
0
else
Determine the value of V ar(2X − 3Y + 8).
[14] DeGroot, section 4.6 Suppose that X, Y and Z are three random variables such that V ar(X) = 1,
V ar(Y ) = 4 and V ar(Z) = 8. Suppose also that Cov(X, Y ) = 1, Cov(X, Z) = −1 and Cov(Y, Z) = 2.
Determine
1
1. V ar(X + Y + Z),
2. V ar(3X − Y − 2Z + 1).
[17] DeGroot, section 4.6
Let X and Y be random variables with finite variance. Prove that |ρ(X, Y )| = 1 implies that there exists
constants a, b and c such that aX + bY = c with probability 1.
2